If you plan on owning a long gun with a barrel shorter than 16 inches, you will have to comply with the NFA laws. This requires you to submit a form 5320.1 commonly referred to as a "form 1." If you apply with a gun trust, there is no identifying information for the BATFE to run a background check. Once the form is approved, they will mail it directly back to you. Once you have the approved form in hand you are legally permitted to purchase/install a short barrel on your firearm.

A form 1 with a trust is the only NFA process which does not require any type of background check or even a standard form 4473 which is needed to take ownership of any NFA item after form 4 approval.