You can certainly refuse service for any unvaccinated dog. Moreover, it may actually be against the law if the dog hasn't been vaccinated against rabies. They can opt out of any of the other shots, but not rabies. They may be able to get fewer shots over the years because holistic vets will do titer tests to check the level of antibodies already existing in the dog's system and then provide documentation to health authorities that the dog is still safe.
From Rover's Terms of Service:
2.2 Rover.com does not provide Pet Care Services....Pet Owners and Service Providers are solely responsible for making decisions that are in the best interests of themselves and their pets. For example, each user of the Rover Service is responsible for keeping current his or her own pet’s vaccinations, and we will have no liability for anyone’s failure to vaccinate his or her pet.
2.10
3. Eligibility; Legal Compliance. By accessing and using the Rover Service, you certify that you: (1) are 18 years of age or older, and (2) will comply with all laws and ordinances applicable to your activities conducted through Rover.com. For Pet Owners, this means, among other things, that you will ensure that your pets are vaccinated and licensed as required by local law.