There are only 23 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.
None of the other races in 5e are at all ooze-like. Technically, nor is there any way tohowever, you can play as a monstermonster—such as an ooze—by playing under the sidekick rules in this editionTasha’s Cauldron of Everything, as detailed in @stevenjackson121’s fine answer.