How? If X people made appointments in a day, then exactly X vaccines should be distributed for that day. Leftover vaccines will only be distributed after all those appointments are over, so it is impossible for any of the X people(assuming they arrived on time) to lose their vaccine.
For instance, one simply needs documented evidence that 1) the right populations were clearly made aware of vaccine location and time, 2) communications went out to "in-scope" target audiences at least X hours before the doses expired, and 3) if vaccines are at threat of expiring once they are - let's say - two hours from expiring then those that control that particular supply should do everything they can to use them.
And while this solution is a bit reductive, it's honestly not that reductive; this is a solvable problem. People on Hacker News solve much, much, much harder problems than this.
And I'd rather 20% of vaccines go to the wrong people than 5% of vaccines go in the trashbin, personally.