Enterprise Alert does have the ability to notify users of gaps in on-call schedules. This option can be set in the web portal.
- Navigate to On Call > On Call Schedules > Teams with On Call
- Click on the team you wish to be notified of schedule gaps
- Click on the Options button
- Check the Notify me daily on schedule gaps for the upcoming days (input the number of days you wish to be notified in advance of schedule gaps)
- This setting is set per user per schedule, therefor this needs to be set for every user and every schedule that they wish to be notified on.