The number is disconnected and the address provided is to the property I know is vacant (I see it every day out my window, trust me, it is vacant).

They'll tell us they had a stack of letters and the majority of people didn't return voicemails, flat out didn't show up to appointments, and they called a lot of disconnected phone numbers.

How do I locate a contact for this vacant property if the property owners appear as deceased, phone numbers are no good/disconnected, what's next on my research?

I agree central a/c is the way to go.But you will need the a/c condenser, a pad (or suspend it on brackets), evap coil, line set, electrical circuit & disconnect so you're probably looking at closer to $2500-$3000  using compatible Goodman parts & depending on what they charge to install it, (usually around $800-$1000).

I think there's a disconnect between some degrees and the real world.

Yes, you can disconnect.

@Joanne Tsai I'd likely ask the agent about the disconnect. 

Propane Fuel Tanks- disconnect and store if leaving for long term storage.J.

My realtor noticed that a pipe on the water tank was disconnected on one end and said that the house would flood because of it if the water was turned on (the house has also been winterized).

From the exit, you take five steps up, and you're outside.  3: The basement does include a full kitchen and 3/4 bath, although the seller disconnected the stove and put it in a closet for closing, for reasons that I didn't fully grasp.