You can customize Amion to set up online preference calendars with checkboxes only for days or shifts that have not been filled.

Start OnCall and go to File / Preferences / Amion requests. Turn on the checkbox that says "Once a schedule is published, show only unfilled slots in online preference / request calendars".

Next go to File / Publish on the web. Advance the publish-to date so that any month where you need people to fill remaining shifts is published. If you want people to submit availability for a handful of open shifts in March, for example, make sure you've published the schedule through at least March 31.

When people go to their preference calendars for published months, they'll see checkboxes only on days that have unfilled shifts. They'll also see whatever shifts they've already been assigned, just as they would in their personal calendars.