I believe that an ultrafinitist arithmetic would still be incomplete. By that I mean that classical mathematics could prove that a sufficiently powerful ultrafinitist arithmetic is necessarily incomplete. The exact definition of "sufficiently powerful", and more importantly, the exact definition of "ultrafinitistic" would require attention. I'm not aware of any such result or on-going investigation.

The possibility of an ultrafinitist proof of Gödel's theorem is a different question. For some definition of "ultrafinitistic", even the well-known proofs of Gödel's theorem qualify. Mayhap^1 someone will succed where Nelson failed, and prove that "powerful systems of arithmetic are inconsistent". However, compared to that, Gödel's 1st incompleteness theorem, which merely states that "powerful systems of arithmetic are either incomplete or inconsistent", would seem rather... benign.

^1 very unlikely, but not cosmically unlikely