Notify the sellers and if the lease you are inheriting indeed says something to the effect of “2 installed working air conditioners” are the responsibility of the landlord then have them do that before closing. 

If your allergy causes your eyes to water and your entire contact with the unit is the 5 minute annual property check, it would be a reasonable accommodation to allow the animal.If your allergy causes your throat to close and stops your ability to breathe and your contact with the unit is picking up the rent in person every month, spending hours inside the unit doing all the maintenance, replacing the air filters every month, quarterly property checks, and the person takes their ESA dog in and out of the building using the only hallway and prevents you from entering other units, it may be reasonable to not allow the accommodation. 

With everything going on in the world right now, a story like yours is a welcome breath of fresh air through hardships.

Also, exit strategy is either Air BNB (property is on a lake), live here permanently as our primary residence after some rehab (residence is livable right now), BRRRR, or house-hack and sell in a few years. 

Including the air quotes because zoning hasn't given their final blessing that it's legally conforming.

@Andy Camarda - I've used DC Air Heating and Cooling at the referral of my GC on several Northside jobs this year. 

Im just a plumber but know about electric from doing air conditioning. hope that helps man

Central air replacement is costly compared to ductless AC.

Missing transition pieces throughout the home, air register covers rusty/don't completely cover opening7.

If you were adding forced air and all new duct work would be upwards of $12K.3.