Yes. There are virus vector vaccines already in existence: https://en.wikipedia.org/wiki/Viral_vector_vaccine
For example, adenovirus vector vaccines were created for COVID-19: https://en.wikipedia.org/wiki/COVID-19_vaccine#Adenovirus_vector_vaccines
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: https://en.wikipedia.org/wiki/Attenuated_vaccine
Oral polio vaccine is a well-known example.
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