Both vaccinated and unvaccinated persons can be COVID infected and infectious.

Vaccination assists the immune system to better fight the infection. That is all.

So a no-fly list would need to pertain to those who are not verifiably COVID negative. This is especially critical in the case of the Delta strain which is far more infectious, even with vaccinated persons, than the earlier strains. But we also have to consider that tests have been shown to give both false positives and false negatives. So the risk cannot be totally eliminated.