For a few months post infection they are probably golden, but longer than that, doubtful. Lots of people have been infected more than once. The most recent example in the news lately was Sarah Palin. The most recent example among my friends, (and not in the news), my friend E was infected in April of 2020, had several months of long covid, has since been vaxxed and boosted, and then caught it again (likely omicron based on the symptoms) about a month ago.
Vaccination is not some magic forcefield. A virus that mainly enters the body through the nose is likely going to be able to at least get a brief foothold in the nasal cavity before the body's defenses can do their thing.