As far as I know a trustee can purchase and pick up items from a dealer on behalf of the trust. The trustee would fill out the 4473 for the item(s) at the dealer.
ETA: I've had a trustee pick up a suppressor after Form 4 approval.
A trustee can act on behalf of the trust, both buying and selling items. That said, make note that a successor trustee CANNOT act on behalf of the trust until they become a trustee.