There are only 3 options for actually, officially, having the “ooze” creature type in, at the very least, all of Wizards of the Coast D&D: the 5e plasmoid race, the 5e sidekick option, and the 3e oozemaster prestige class.
There are a number of other options for being something “ooze-like,” all in 3e and the “v.3.5 revised edition.”
Prior to 3e, we get into TSR D&D, and I don’t know it as well. I haven’t been able to find anything, but that doesn’t mean it isn’t out there.
Plasmoids, from Spelljammer: Adventures in Space, are the only actual honest-to-god playable ooze race in D&D history, that I can find. In addition to being “amorphous” and without “internal organs of the usual sort,” their trait list outright says:
Creature Type. You are an Ooze.
None of the other races in 5e are at all ooze-like. Technically, however, you can play as a monster—such as an ooze—by playing under the sidekick rules in Tasha’s Cauldron of Everything, as detailed in @stevenjackson121’s fine answer.
Every playable race in 4e:
Human, Half-Elf, Halfling, Dwarf, Elf, Eladrin, Dragonborn, Tiefling, Deva, Half-Orc, Longtooth Shifter, Razorclaw Shifter, Goliath, Gnome, Minotaur, Githzerai, Shardmind, Wilden, Drow, Genasi, Kalashtar, Changeling, Warforged, Mul, Thri-Kreen, Gnoll, Revenant, Shadar-Kai.
None are remotely ooze-like. D&D 4e also provided no way to play as a monster (and it would be really, really hard to homebrew one, since PCs and NPCs operated off of entirely separate rules).
D&D “v.3.5 revised edition” is by-far the edition I know best, so here’s what can be found in that edition.
Ooze-like officially-playable races
The ghaunadan are a race of evil ooze-like shapechangers devoted to Ghaunadaur, and in 3.5e they were technically playable, though in practice I don’t think it would have been plausible.1 However, they were not technically oozes—their originally incarnation in Monsters of Faerûn had them as “shapechangers” which isn’t a thing that can stand on its own in 3e (it’s a “subtype”), and the errata to Player’s Guide to Faerûn (which officially was the last word on updating a lot of things to the revised edition) states they are aberrations (the same type used by illithids, beholders, and the like).
Also in Faerûn, we have the Underdark book2 and its slyth race. The slyth are regular humanoids, but they do have the ability to turn into an ooze-like “amorphous form” for up to 10 × level minutes per day, broken up as the slyth sees fit. That form is pretty much solely for infiltration, though, since the slyth isn’t really allowed to do much while using it except move around (including through incredibly tight spaces). Anyway, it is officially, technically, playable, though again I suspect most would find the experience frustrating.3
Intelligent oozes, including gelatinous cubes—not officially playable
One of the hard, practical stops on playing any monster is Intelligence—even if you ignore the rules saying you’re not allowed to play one, you can’t really turn a mindless creature into a player character. Most oozes—including the gelatinous cube—are mindless, but D&D 3.5e does provide ways around that.
For example, there’s the awaken ooze spell (Dragon vol. 304), or the sentry ooze template (Dungeonscape), though the latter only provides “animal-like” Int 2. There are other templates that are not specific to oozes, but still legally applied to one, that provide more Intelligence. Ultimately, however, none of these rules officially make the cube playable.
Wizards used those same ideas in its own “Elite Opponents” article that includes a gelatinous cube monk, but it’s not playable either—monsters could take class levels in this edition. It gets around the gelatinous cube’s lack of Intelligence by using the fiendish creature template. They did similar things with other oozes in another “Elite Opponents” article.
There are also various oozes that are intelligent by default, mostly in Monster Manual II and III. These aren’t playable either.
Ooze-related class features, including becoming an ooze
Clerics could have the ooze domain (Fiendish Codex I: Hordes of the Abyss) and/or the slime domain (Player’s Guide to Faerûn), allowing them to rebuke and command oozes they found while adventuring. Such a cleric could build up an ooze army. Both domains offer this same feature, but have different associated spells and in any event you could take both to get more uses per day. And maybe you’d be able to use the two command abilities separately, effectively allowing you to command twice as many oozes? I don’t believe there’s an official rule on that.
Sorcerers and wizards can try to do something similar by using the 6th-level spell ooze puppet (Spell Compendium). That doesn’t appear to have any cap on how many puppets you can have at once, beyond your available spell slots. Considering that each casting lasts a minimum of 11 days, that could be a lot. On the other hand, as an 11th-level character, many of your enemies are probably not going to be too scared by most oozes, even in large numbers.
Speaking of spells, there are several other relevant and related spells:
Mold touch (Player’s Guide to Faerûn) is a 3rd-level spell available through the Initiate of Nature feat. It summons brown mold on the spot you touch, which can be another creature. Brown mold isn’t a creature in 3.5e, so it doesn’t have a type, and might arguably be more of a fungi thing than a slime thing. Then again, there’s overlap there.
Engulfing terror (Drow of the Underdark4) summons a gelatinous ooze as a 3rd-level druid, sorcerer, or wizard spell.
Amorphous form (Spell Compendium) is a 3rd-level spell for assassins, sorcerers, and wizards that allows you to take on many ooze traits temporarily. Doesn’t technically make you an ooze though.
Touch of Juiblex (Book of Vile Darkness) is an absurd 3rd-level “corrupt” spell that turns its target into green slime over 4 rounds. A corrupt spell can be used by anybody willing to pay the price, in this case taking 1d6 Strength damage. Anyway, during those 4 rounds, you need remove curse, polymorph other (a 3e spell that was removed in the 3.5e revision), heal, greater restoration, limited wish, miracle, or wish to undo it. After the 4 rounds, there doesn’t seem to be anything you can do about it, except I guess destroying the green slime and casting true resurrection. Maybe resurrection on the slime itself? Regardless, those are high-level spells that a lot of targets are not going to have handy, particularly not at 5th level, making this a monstrously effective tactic. I’d ban it, to be honest.
Mantle of the Slime Lord (Champions of Ruin) is a 7th-level spell available via the Initiate of Ghaunadaur feat. It gives you some ooze traits and makes mindless oozes ignore you.
Simbul’s skeletal deliquescence (Magic of Faerûn) is an 8th-level sorcerer or wizard spell that “is normally used as a punishment or to disable opponents without killing them,” by turning their bones to “mush” and making them “oozelike.” It actually does provide the target some of the benefits of being an ooze, but since it doesn’t provide the target with any ability to move without their bones, this is small comfort.
Wall of ooze (Book of Vile Darkness) is a 5th-level spell for clerics, sorcerers, and wizards that does what it says. The wall isn’t a creature, per se, nor is it mobile or anything, but anyone who touches it does take acid damage and risk paralysis. If they are paralyzed, the wall tries to digest them, and adds their hp to its own if successful.
Yochlol blessing (Drow of the Underdark4) is a 4th-level cleric spell that turns someone into a yochlol. Yochlols are kinda oozy.
D&D 3.5e was also well-known for its panoply of “prestige classes,” classes you couldn’t start out in but had to multiclass into after meeting the class’s requirements. Several are relevant here:
Slime lord (Player’s Guide to Faerûn), another thing devoted to Ghaunadaur, becomes very ooze-like themselves, gaining most of the features associated with oozes over 10 levels. They do not, however, technically change to the ooze type.
Thralls of Juiblex (Book of Vile Darkness) get a lot of features similar to the slime lord—many ooze traits, but no official ooze type.
Oozemasters (Masters of the Wild) actually do turn into oozes, technically and officially. Weirdly, though, they don’t really get any kind of “mastery” over oozes—even if you were a slime-domain cleric who multiclasses into oozemaster, it won’t progress your ability to command slimes, so you’ll be really limited on that front. It does progress your spellcasting at most levels,5 though, so I guess you could use spells to summon oozes.
Spell sovereigns (Dragon vol. 357) are all about living spells, which are just that, spells turned into creatures. Living spells are oozes under 3.5e rules, so technically this class is about (one specific kind of) oozes. But not oozes in general, and in any event the spell sovereign only controls living spells; they don’t become one. On the other hand, it’s also definitely the most powerful way to specialize in oozes, since living spells are incredibly flexible and the spell sovereign gets one as a familiar, which opens up all kinds of familiar-related class features, feats, and spells. That includes options like the Extra Familiar feat (Dragon vol. 280) to get more than one, and a changeling wizard’s morphic familiar feature (Races of Eberron), to swap around which spells each of your familiars is. On top of that, it advances your existing spellcasting at most levels.5 (Credit here to @forrestfire, who is playing a character similar to what I’ve described here in a game we’re in and worked out this approach.)
Master of many forms (Complete Adventurer) is a class devoted to, well, mastering many forms, expanding on a druid’s wild shape and getting into all kinds of creatures—including oozes at 8th level. They can actually become an ooze, but only temporarily. It lasts a long time and they can use it a lot because they’re heavily specialized in it, so they probably could stay in ooze form all day, every day if they really wanted, but it’s not what the class is for.
D&D 3.5e used a “Level Adjustment” system, where a character of a “powerful” race like ghaunadan counts as a higher level than they actually are. A ghaunadan’s is +5, so a 1st-level ghaunadan is supposed to count as a 6th-level character. The benefits of being a ghaunadan do not—remotely—compare with the benefits of five class levels, making any ghaunadan character feel very much “behind” their nominal peers—in fact, by comparison, the ghaunadan will most likely feel utterly crippled. A 1st-level ghaunadan is simply not up for dealing with 6th-level threats.
Not to be confused with Drow of the Underdark, which is a setting-agnostic supplement with similar themes from the same edition.
Like ghaunadans, slyth had an LA. It was only +2 instead of +5, but again nothing about the slyth was even remotely worth two class levels. It might just barely be something some masochist was willing to put themselves through, but I cannot more strongly recommend against it.
Not to be confused with just Underdark, the Forgotten Realms supplement mentioned earlier.
Note, however, that missing even one level of spellcasting progression is widely considered near-crippling in D&D 3.5e. Spells are just that powerful. On the other hand, spellcasters are so powerful they can afford to cripple themselves a little.
I have absolutely no idea; you’ll need an answer from someone well-versed in those systems for those.
Pathfinder third-party material
This isn’t really D&D, since it wasn’t published by D&D’s owners or even for D&D itself, and for that matter, it’s not even really Pathfinder, since it wasn’t published by Paizo. But it’s also exactly what you’re looking for, and Pathfinder is kinda-sorta D&D.
There is a “pay what you want” third-party publication by Dreamscarred Press, April Augmented, that includes a playable gelatinous cube. Converting Pathfinder material to D&D 3.5e material (which it’s based on) is not hard (but does also require some effort, you can’t just use it as-is).
Disclaimer: That’s April Fools’ material. DSP took pride in even their joke material, and all of it is intended to be usable even in serious games, but it’s worth noting.
Disclaimer the second: I contributed to that book. I didn’t work on the playable gelatinous cube, and it was a volunteer gig in the first place, but you should be aware that I have a vested interest in this publication.