Not a shot, an intranasal spray. The leading candidates on clinicaltrials.gov are adenovirus carried spike protein based versions (I would rather have mRNA) but it should still do the job of getting upper respiratory resident T cells trained on sars-cov-2 epitopes.
The igA antibodies produced by trained mucosal resident B->plasma cells would stick around for an even shorter time than the igG antibodies do in the body serum. But the trained and resident T cells would be there to kill off infected cells right at the start instead of after days. So yeah, I think a sterilizing intranasal vaccine is possible but it likely won't depend on the igA antibodies generated. This follows with the studies of "super" immunity in rare special medical workers who never get sick with covid-19 and also never show antibodies for it. The nature paper seemed to show it was their upper respiratory mucosal T cells doing the job; having already been trained and cross reactive with other coronaviruses. But we can't all be lucky mutants and most of our upper respiratory systems will need some training.
