The closest to getting soft serve for a party would be Carvels. They don't have a cart, but they do sell their soft-serve cones pre-dipped in the freezer. There are four to a box. I don't know how the pre-made cones taste as I buy fresh from the store.
In one way that might not be a bad option. If you had some sort of cart that you rented to keep them frozen, it would avoid people standing on line waiting for a cone and be less drippy and messy. They have other frozen novelties too like flying saucers, sprinkle cups, chipsters. If interested the store is in Berkeley and another one on the Pennisula somewhere.
The store doesn't carry all the stuff mentioned on the site, but maybe they can be pre-ordered.
Recently there was a mention of Silberman's which has a cart but no soft serve
Given the size of the crowd, maybe an option would be to have a few vendors ... one gelato cart, one ice cream cart ... Melanie has been mentioning some lemon ice vendor.
Contact Berkeley Farms. They may rent you a party cart that's stocked with 3-gallon buckets or novelties (including Its-Its!). As for soft-serve, the equipment's pretty heavy & expensive, so I'm not sure if anyone rents them out. Try a party-supply store like Diddam's - they rent cotton candy machines, rolling hot dog grills, etc.
Also, try this: