And the lease goes until May 2021, so hopefully we'll have some sort of vaccine or drug by then...

Landlords Can Require COVID Vaccines"Whether your landlord can require proof of a COVID-19 vaccine depends on your state of residence.

The landlord may also require a "pet health certificate" to give him/her assurance that the pet is in general good health, has received all state required vaccinations, and is parasite-free.

For example, you absolutely can require the animal have a current veterinarians health report and current vaccines.

Pretty sure that, regarding the service dog's "authenticity," you can only ask for written verification from the tenant's health provider that the service animal is necessary, in addition to the animal's vet records to show proof of vaccination status and in good health.

Can I require the dog to have current vaccinations? 

US rallied and Linus Pauling in Palo Alto created a vaccine..

I would require prospective tenants provide pet vaccine records as part of the application process.

Must be able to provide vaccinations records, non aggressive breed, are they ok with paying higher sec dep and having $25 fee per month for having pet?

Luckily I have not run into the issue yet but I plan on directing tenants to petscreening.com to check vaccination records, possibly known history (landlords can document pet history for future landlords), and legally validating if the animal is an assistance animal.