Can I dismiss an employee if they refuse to have the vaccine?
See above FAQ about whether you can demand that your employee has the vaccine.
Dismissal for failing to follow a reasonable instruction would be a possibility but it should be the last resort.
First you will need to be able to show that you have reasonable grounds for insisting that they have the vaccine. You will then need to demonstrate that you have taken into consideration the reasons why the employee has refused and why they are not considered reasonable. Before taking a decision to dismiss you should look at alternatives such as other duties/other roles.