short barrelled shotguns & rifles fall under the category of "other" and the transfer tax is quite a bit less than the the tax on transfer of machineguns. Machineguns & short barrelled rifles and shotguns are not prohibited, only taxed. In the process of taxing you, the JBT's at ATF conduct a background check on you.
Ah, I'd heard it fell under Dept of Treasury, so that makes sense. But I was under the impression you had to get the permit from them, which would be like asking San Francisco to issue a concealed carry permit.