I work at
Disneyland Resort, and I don't know how things work at WDW.
I keep complete costumes at home for the two locations where I usually work. If I am spending the whole day at work, I come dressed and leave dressed. I usually drop off (at costuming) the shirt I wore yesterday, and pick up a clean one for tomorrow as I am leaving at the end of my shift. I often stop at the market, or Starbucks etc. on the way to or from work while in costume, but without my name tag - the name tags are not supposed to be worn when you are off Disneyland Resort property, but you are allowed brief stops at vendors going to and from work in costume. Extended wear (like going to the movies or a ball game) is pretty clearly outside the guidelines.
If I am working at a different location, or if I am coming in as a guest early and working a later shift, I bring my shoes, name tag, belt, tie, hat, jacket if needed, and leave them in a locker at costuming, then later return and check out any other needed costume pieces and dress at costuming before starting work. If I have an early shift and am returning to the park later as a guest, I come dressed and bring my "civilian" clothes with me, leaving them in a locker at costuming for later use.
The 800 costumes mentioned is probably 800 distinct roles. For each role, there are many costume pieces in each needed size, so the actual number of costume pieces at Costuming is enormous. Some overlap - for example, the dark blue pants and black belt for men are the same for Blue Bayou and Carnation Cafe. Each costume piece has a bar code label, which is scanned when the piece is checked out to you, and scanned when the piece is returned to Costuming.
A rather amazing operation!