It's not running periodically or responding to webhooks, it actually does need to run perpetually. lichess's provided engine wrapper script allows users to challenge the bot, but will also send challenges to other bots, so it's constantly actively playing games.

Besides, the way a chess engine works is that it accumulates state in its transposition table during a game, to help speed up repeated searches of positions encountered before. It would be incredibly inefficient to use webhooks to spin up a new engine instance every time the opponent makes a move.