This is one of Vox's Future Perfect predictions for 2023; they give it a 90% chance:

For a long time, we’ve been hearing about how Covid-19 vaccines delivered through the nose would likely prevent more infections than shots in arms. And China, India, Russia, and Iran have already greenlit vaccines taken through the nose or mouth. Alas, not the US. Nasal vaccines created by American researchers have been tested in animals, but human testing has been held back for a few reasons. A big one is the lack of funding: Biden has asked Congress for more money for next-generation vaccines, but Republicans have resisted. Current estimates put nasal vaccines years away for the US. That’s depressing, but the indications suggest it’s accurate.

(Vox)

Resolves according to Vox Future Perfect's judgment at the end of the year.