In Claude Code on the web (code.claude.ai), you can set up a scheduled session that runs on a cron schedule. The session starts a fresh Claude agent, which reads its instructions and calls whoami + get_my_tasks then works the next task in the queue.
Give it a prompt like: "Call whoami, then get_my_tasks — if there's a task to claim, claim it and do the work. When done, set status to done."