Depends on the state but most states will just slap a temp tag on it and have you drive it home. Depending on the dealer you may pay the price of the car but no sales tax, then pay the sales tax when you go to the DMV and register it in NY. If it’s a bigger dealership they may handle everything for you, which is what Audi Atlanta did for me. I flew in, car was already tagged with a GA temp tag and ready to roll. I wouldn’t have done the deal without seeing the car except it was a full CPO that I knew if I found any mechanical issues later on I’d be able to get them fixed at any Audi dealer. They handled all the NY paperwork and my plates came in the mail a couple weeks later. The Enclave that I got in PA was mostly the same but that wasn’t a done deal before I arrived so it wasn’t pre-plated. We finalized the deal and my trade then I went across the street to a BBQ joint/gun shop (man I love PA sometimes) and by the time I came back the temp tag was on it and I drove home. Again plates came in the mail. The GTO I got in NC and in that case I paid for the car with no sales tax, got a temp tag, and drove it home. Then I had to go to the DMV and pay the sales tax and get my plates/registration.
I think some states you’ll still have to pay their sales tax, then depending on the state you can claim the sales tax you already paid as a credit toward the NY sales tax. Be careful here because the list of states without a reciprocal agreement is pretty long and you don’t want to end up paying twice. I have never run into this and I’ve bought cars from dealers in PA, OH, GA, MI, and NC. In college I even bought a car private sale in Kingston, Ontario and brought it home on a trailer. That’s a whole lot more complicated with DOT and EPA getting involved. Page 2 of this DMV form goes over the simple case of buying from other states:
One last tip… if the temp tag is cardboard, and they almost all are, don’t let them attach it like a plate if there is any chance of rain. My temp tag on the Enclave got soaked on our way home and came off somewhere between Binghamton and home, leaving me with a car with no plates. The PA rules did not allow the dealer to order a replacement temp tag and the paperwork was already submitted for NY plates so the NY DMV couldn’t do anything. The dealer ended up overnighting me a set of dealer plates to drive around on for the two weeks it took NY to get me the damn plates.