You can have your trust "own" the gun, but the 4473 would be filled out by you the individual trustee and you still have to abide by the rules.
18 USC 922
(b) It shall be unlawful for any licensed importer, licensed manufacturer, licensed dealer, or licensed collector to sell or deliver—
(1) any firearm or ammunition to any individual who the licensee knows or has reasonable cause to believe is less than eighteen years of age, and, if the firearm, or ammunition is other than a shotgun or rifle, or ammunition for a shotgun or rifle, to any individual who the licensee knows or has reasonable cause to believe is less than twenty-one years of age;
It's no different than when an LLC or a corporation buys firearms, they're the property of the LLC/corporation but they still have to send an individual to pick them up and the dealer still has to make sure that person isn't prohibited.