I honestly don't think you have much of a choice in this arena. Probably the only camera that's going to fit the "instant photo"/polaroid bit of that that you can easily obtain new for a lower-cost is going to be a Fuji Instax Mini camera--and the prints made are tiny--roughly business-card size. The problem, budget-wise, is going to be the film, and the quality of the images is probably not what you'd like.

You may want to consider getting a Fuji Instax SP-1 printer, and letting everybody use the Instax Share app on their smartphones instead, so they can also have a digital image of higher quality as well as the print (and you typically don't need to teach people how to use the camera in their phone), but you're still going to run bang on into the cost of the film/batteries (it uses a weird CR2 battery, although you can also use smartphone external battery packs with a special cable).

And no, these Instax cameras can only do 10 at a time--that's how many sheets are in the cartridge you load into the camera/printer.

Your only other option, aside from vintage restored Polaroid cameras, is liable to be the Impossible Project's Instant Lab, but that's pretty close to all of your budget without throwing in the fim.