Yes, that is permitted. The employer may ask his employees if they have been vaccinated. There must be good cause for this. Good cause implies that the request must be proportionate and necessary, and there must be no alternatives. The employee does not have to answer the employer's question.
The employer may not record the vaccination status (i.e. whether or not you have been vaccinated) of individual employees in any way. Under the General Data Protection Regulation (GDPR), data concerning health is designated as special personal data. The employer cannot process special personal data unless there is a legal exception.