No emails are sent to users
Emails are sent via a SharePoint workflow. This is also responsible for auto-approving leave requests that have Requires Approval set to false.
If no emails are sent and/or approval for certain requests are not working, verify that workflows are not blocked in your enviroment.
Verify that workflows are not disabled/blocked by making sure the checkbox for Block SharePoint 2013 workflows is unchecked by going to https://***-admin.sharepoint.com/_layouts/15/online/TenantSettings.aspx where *** is the name of your SharePoint domain.
Another issue may be that users are lacking permissions to the underlying workflow lists. If users by default only had read permissions, they need to explicitly have edit/contribute permissions to the LeaveCalendar/Lists/WorkflowHistoryList and LeaveCalendar/Lists/WorkflowTaskList. You can change this buy going to these lists, click Modify View under the List tab in the ribbon menu, click Settings in the breadcrumb title of the Modify View page and finally clicking List Permissions from the settings page of each list.