Can I use OnMe on any Minecraft server?

Only where your use is authorized. OnMe doesn't vet individual servers, so it's on you to check a server's own rules on automation, AFK play, and multi-accounting before pointing a bot session at it — some servers restrict or ban this kind of activity outright, others allow it, and OnMe has no way to know which is true for a given server at a given time.

Separately from server rules, you're responsible for complying with the Minecraft End User License Agreement, Minecraft Usage Guidelines, Microsoft Services Agreement, and Xbox Community Standards for every connected account. None of that responsibility transfers to OnMe — automated or AFK play may violate a platform's own policies even on a server that technically permits it, and any consequence to a connected account (suspension, ban, or otherwise) is yours to deal with.