In order for an employee to be able to add sickness from their Self Service page, they will need the add and edit permissions for 'Sickness Selfservice'.
Go to Setup and click Roles & Permissions.
Click the Employee role.
Search for 'Sickness Selfservice'.
Click the plus and pencil icons for this permission so that they are set to green.
Click Save.
Next, go to the affected employee's profile and click Security.
Click Roles and make sure the Employee role is highlighted in blue.