Or is that in addition to your emergency fund?

With that being said if I put 3.5% down with a FHA loan that would be $10,500, plus 2% closing fees ($6,000) and I estimate a 3 month emergency fund to be around $7,500 on the safer end.

keep one for emergencies ... 

They should only knock on your door when it's an emergency.

. $21.55 billion for emergency rental assistance through September 30, 2027$5 billion in emergency housing vouchers through September 30, 2030$100 million for tribal housing improvements$100 million for rural housing through September 30, 2022$5 billion to assist people experiencing homelessnessThe added $21.55 billion has increased the overall assistance to $45 billion from Congress for renters.

They can close extremely quick and can serve as your emergency plan or even plan a if their offer is high enough. 

I have great tenants who always pay on time but just had an emergency this month and rent is late.

I currently keep my emergency fund in a ROTH.

As long as the cash flow and other due diligence numbers make sense, you have a plan for property management and some backup cash for emergencies, it should be doable.

It is not your fault your tenant did not save cash to prepare for emergency situations.