Would it be safe to take my unvaccinated house cats on walks in a stroller?

I'm thinking of buying a double decker cat stroller to take my cats with my puppy and I on walks. For health purposes they have been unvaccinated since the kitten shot series. They are in their late teens, but still quite vigorous.. I'd think they'd have encountered any germs they'd get from the air already since my puppy comes in from walks and plays with one of them. I would avoid all other animals and people at all costs