Our tenants have one dog who seems well behaved and with an up to date vaccination record.

You'll want to read this article:  https://www.thebalancesmb.com/is-a-landlord-liable-if-a-tenant-s-dog-bites-someone-2125072Without exception, you will want to require your tenants to carry Renter's Insurance and have a strong pet addendum - including a photo of the pet, breed (non-aggressive only), the pet's medical records to show all vaccinations are current, more...And, if a dog bites anyone (other than someone breaking in or the like), you will need to act to have the pet removed from the premises to ensure you do not assume liability.

I have a strict pet policy -- a well-trained dog, over 1 year (no puppies for sure), breed restrictions per my insurance, vet records showing age, breed, current vaccinations, required city license, and pet interview with landlord (me) or K9 good citizen certificate. 

It requires the tenant to upload a picture of the animal, vaccination records, etc.

I permit small dogs with breed restrictions, proof of vaccinations, a photo and insurance, with a pet fee. 

I included the restrictions from my insurance, the requirements of spay/neuter, vaccinated and registered with the City of Chicago, older than 15 months, and that a vet certify the breed (as I learned here that some tenants lie and say Labrador, when the animal is actually a pitt mix, or another breed that violates home insurance.) 

I don't discriminate, I screen the animal, require registration - which implies current vaccinations or titer testing, spay/neuter, and I get a sense for the temperament.

Should I ask for vaccination or/and registration of pets?

I say Yea (full disclosure - pit bull owner/landlord/realtor) but only if:Tenants pay a fee.You see copies of vet records showing vaccines, annual visits, etc.You see rabies certificate.You see dog license.You see training certificate/get recommendation from vet/trainer.They agree to pay me ($25) for each time my landscapers have to pick up dog doo.Bonus points for them taking dog to day care 1-2x/week or getting extra liability insurance.If they are willing to jump through these hoops, they are likely responsible owners and worthy of renting to.

.- His charm is so contagious, vaccines we’re created for it.- Years ago, he created a city out of blocks.