Yes, you can ship your firearm to Gunsite for your class. Be sure to address it to yourself c/o Gunsite Academy. Since FedEx and UPS will only allow firearm shipments from FFL to FFL, you will have to ship through your local gun store. You will be able to acquire your firearm once you arrive for class and we will ship it back to that same FFL after your class, at your expense.
You might look at the TSA regulations on flying with a gun. It isn’t that onerous, is simple and adds just a little bit of time to your travel. (I do it regularly.)
https://www.tsa.gov/travel/transporting-firearms-and-ammunition
Please call 928.636.4565 ext. 2 if you have questions.