We do allow a well-trained dog, preferably large, no puppies and no breeds not covered by our insurance (vet records showing breed, age, proof of current vaccination and any required pet license).

I now require any animals brought onto the property for any length of time, even service and emotional support animals, to be registered with petscreening.com or otherwise verified through a third party, and if tenants or their guests are not happy with that, they can send the documentation directly to me (vaccination records for the animal and a doctor's note with medical license and clinician contact information be sent to me).

I would also request a VERY detailed background history of the dogs - most people who are trying to "beat the system" don't go the whole 9 yards of building a credible vaccine / vet check ups report.

Your Animal agreement should cover licensing, vaccinations, a photo of the pet, flea control, instructions on waste disposal and an inspection schedule. 

It is also recommended that the LL have an ANIMAL -- not pet -- policy covering care, animal waste, vaccinations, insurance, inspections and damages.I would inspect and immediately charge for damages. 

@Tim Bartel any cat or dog over the age of 1 year is considered an adult and at that point several behavioral problems have been corrected and they should largely be up to date with pet vaccinations.

Still waiting for that vaccine to hit the market.

I attach this to the lease and it covers all aspects of owing a pet including vaccinations etc.

Require them to provide documentation that their animal is licensed, vaccinated, and flee free2.

Make sure it's spayed or neutered, and registered - which also insures it's up to date on vaccinations.