Does the possessor of an NFA firearm have to show proof of registration?

Yes. The approved application received from ATF serves as evidence of registration of the NFA firearm. This document must be made available upon request of any ATF officer. It is suggested that a photocopy of the approved application be carried by the possessor when the weapon is being transported.

[26 U.S.C. 5841(e); 27 CFR 478.101]

Source ATF: https://www.atf.gov/firearms/qa/does-possessor-nfa-firearm-have-show-proof-registration