The Form 4 should be submitted with supporting documentation that the trust is legitimate. They'll check that.
Upon approval, the actual person picking up the firearm from a dealer will usually be subject to a NICS check. This is something of a gray area. According to the NFA handbook:
9.12.1 NFA Transfers to other than individuals. Subsequent to the approval of an application requesting to transfer an NFA firearm to, or on behalf of, a partnership, company, association, trust, estate, or corporation, the authorized person picking up the firearm on behalf of, a partnership, company, association, trust, estate, or corporation from the FFL must complete the Form 4473 with his/her personal information and undergo a NICS check. [p.76]
However, the instructions on the 4473 state that no NICS check is required, and there's no reference to this in the GCA. Nonetheless, most dealers will perform one.