The [ScheduledTask API](https://www.w3.org/TR/task-scheduler/) enables the setting of alarms and reminders. It is currently not implemented in any major browsers, maybe because of the obvious potential for abuse. However, the Push API provides the same capabilities now, except every wake-up has to be initiated server-side. So if implemented with similar restrictions, no harm would be done by the ScheduledTask API.
I believe that this API is one of a few missing steps in bridging the gap between PWAs and native apps. Simple applications, like reminder or calendar apps, shouldn't need to be native to work offline.