Can I ask an employee if they have been vaccinated?

You can’t just ask because you want to know.
Vaccination status is “special” category of personal data under GDPR or DPA 2018.

Special information which should only be collected under special circumstances.

Why would you need to now this health information?
To conduct a risk assessment
To assess what is the risk to your workforce?
Or to comply with new test and trace rules – who should be isolating and who doesn’t need to isolate after being contacted by test and trace

You can’t ask just out of interest.
You can collect the information if you have a specific reason.

NOTE – This information must be stored securely