It's almost entirely up to the DM.
Let's start with the assumption that you cast the spell on the platform with the party already standing on it (which is already problematic). Further down, I'll address what happens if you cast the spell when the platform is empty and the party later climbs aboard it (which is even more problematic).
From the text of the levitate spell (emphasis mine):
One creature or loose object of your choice that you can see within range rises vertically, up to 20 feet, and remains suspended there for the duration. The spell can levitate a target that weighs up to 500 pounds. An unwilling creature that succeeds on a Constitution saving throw is unaffected.
The platform must be considered a "loose object." This isn't some defined game term, so we can use a basic English interpretation: it's not attached to anything; it can move freely and independently; it's not combined with anything. If a party is standing on the platform, it's not really loose by my estimation as a DM, because it's in contact with the party, which restricts its movement substantially. A different DM might rule otherwise; for example, the party isn't permanently affixed to the platform, so they aren't really attached in a way that restricts the platform's movement. Regardless, what it means to be a "loose object" is entirely up to the DM.
There is a clear RAW reading of the weight limit: if the platform itself (the "target") weighs up to 500 lbs then it can be levitated. I believe there is a convincing RAI reading as well: if the platform plus anything it is carrying weighs up to 500 lbs (similar to an elevator) then it can be levitated. But, again, whether the platform's weight includes its riders is up to the DM, depending on whether or not they favor RAW readings.
So far, we've assumed the platform was already loaded. If the party isn't already loaded onto it, then the platform may be a valid target for the spell when cast, yet after loading it may no longer seem to be a valid target. Unfortunately, what happens to a spell when cast on a valid target that later becomes an invalid target is ambiguous (see this answer on another question for why). Then even more DM adjudication is required: not only does the DM have to decide the "loose object" and weight limit issues, but also they must decide how to resolve spells whose targets later become invalid.
In any case, if you're the DM, you have to decide away the ambiguities above and you have to decide whether a RAW reading or a RAI reading is more sensible for your game. If you're the player, then you should discuss it with your DM: it's not a great idea to demand it should work on a technicality of the weight limit phrasing, because your DM is not beholden to RAW without their consent. For what it's worth, there's probably no combat benefits to be gained by allowing levitate to be used in this way, and, even though I think the RAI reading is that it shouldn't work, I would absolutely say "yes" to any player who suggested this usage, as long as the environment is suitable, whether the platform is loaded before or after the casting.
