It looks like you can get the permit from servicearizona.com and print it at home without going to the Arizona DMV.
Typically when purchasing a used vehicle or trailer out-of-state, you get the title signed over and a bill of sale, go to the DMV in the state where you're purchasing the vehicle, and get a transit permit (or whatever they call it—a temporary registration of some sort) by showing the paperwork and paying a more or less nominal fee. You then drive/tow the vehicle to your home state and get a permanent title and registration, rendering unto Caesar the things that are Caesar's at that time. Specific details can vary on both ends; state laws aren't entirely consistent. You do need to take care of applicable liability insurance; for a trailer, that's covered by the tow vehicle's insurance policy, I suspect.
If you are a resident of Arizona and the company in Nevada exists only to register your RV, you may well find Arizona getting annoyed at what they believe to be tax dodging. Make sure you comply with all the laws precisely in terms of where the trailer is kept and driven and registered and all that sort of thing.
