No he won't be. It only takes ONE dog to infect your unvaccinated puppy. And even if you get a ride to your vet, DO NOT PUT THE PUPPY ON THE WAITING ROOM FLOOR. Although vets should disinfect their examination table after each patient, I take a rug, or towel with me for my puppy(ies) to sit on.
You could, if you don't drive, get your vet to come out or use a cab? When we had a big litter, our vet came out to do the vaccination/health checks rather than have to take them all in.
And your puppy should not be out of your property (on the ground) for a couple of weeks beyond his final vaccination shots either. It's not much to ask to prevent him picking up one of the sometimes fatal dog diseases surely.
ps I'd also suggest that a 20 min. walk might be too far for some puppies to manage yet, especially back again after having had the vaccination.