I have given Bayesian answers to questions that do not have the bayesian tag. Sometimes my Bayesian answer is accepted and there's not tag to lead anyone there. Is it ever okay to add the tag? Do I even have the authority to do that?
I would usually avoid adding the tag unless there was something about the question that made it Bayesian or at the very least where a Bayesian approach was notably natural or obvious.
Note that adding the tag carries a degree of implication that the OP sought (or should have sought) a Bayesian answer -- if a non-Bayesian approach is just as reasonable, then I would not add the tag, because it could imply other answers (including answers that have not yet been offered) have mistakenly taken a non-Bayesian approach.
Similar considerations would apply when adding a tag implying a particular methodology when some other methodology might quite reasonably be used.
There can be something of a grey area; if you're not sure, one thing you could consider is to ask the OP if they're happy to have the tag there.