We have had vaccines that can spread for a long time. They're called live virus vaccines. Vaccinia is a classic example, as the live virus vaccine for smallpox. My favorite photo from grad school was this lady who touched her boyfriends vaccination site and got the virus in her eyes somehow. Devastating effects. There's several good reasons we don't vaccinate against smallpox anymore, and secondary infections is one of them.

mrna therapy already does this. vaccinated people are shedding cells that produce the protein.