Yes. There are virus vector vaccines already in existence: en.wikipedia.org/wiki/Viral_vector_vaccine

For example, adenovirus vector vaccines were created for COVID-19: en.wikipedia.org/wiki/COVID-19_vaccine#Adenovirus…

Those are non-replicating though.

Live attenuated vaccines also exist – which can replicate – but those work by using a weakened/modified version of the original harmful live virus the vaccine is intended to protect against: en.wikipedia.org/wiki/Attenuated_vaccine

Oral polio vaccine is a well-known example.