Vexatiously is the -ly form of vexatious: it describes HOW something is done — in an annoying way, or, far more often, in a way meant to trouble and harass someone rather than to achieve anything real. This is a rare word. It is much rarer than vexatious itself, and you will almost never hear it spoken. Where it does live is legal writing: 'the proceedings were brought vexatiously', 'he acted vexatiously and without reasonable cause', 'the request was made vexatiously'. In each case the point is the purpose behind the act — the person did it to wear the other side down, not because they had a genuine case. Outside law it can simply mean 'in an annoying way', but even there it sounds heavy and old-fashioned, and a writer would normally choose annoyingly or irritatingly instead. Treat vexatiously as a word to recognise when reading rather than one to reach for in your own writing. If you are ever unsure, use annoyingly for the everyday meaning; for the legal meaning there is no simple substitute, which is exactly why lawyers keep the word.