What kind of a tour is it? Is it a small, private tour or more like RAGBRAI or Bike Virginia? With a private tour company, it's a lot easier to hitch a ride with a support vehicle, although I wouldn't necessarily count on it. Better to ask the tour company your question. If it's a bigger tour, I wouldn't use the SAG vehicle unless you have an emergency or mechanical.

As for walking up a climb and footwear, I suppose it depends on long of a climb it is. I've walked up short steep climbs in my road shoes, but I've never had to go too far in them. I don't generally have any place to stash tennis shoes. Where would you put yours?