Do your same background and credit checks and other requirements that you do with every other applicant. 

from everyone in your current network and make sure that total amount if 150% or more than the money required to close.There are a lot of other factors to consider but the above it a good start.

Although the Fannie Mae product matrix mentions the acceptance of Community Seconds for down payments on Homestyle loans, it's crucial to note that Fannie Mae doesn't directly purchase Community Second loans; they set eligibility requirements for Homestyle loans.

You correctly pointed out real estate investing requires liquidity/cash reserves.

These loans don't require W2's, paystubs, or tax returns.

My tenants (less than 5 units) usually pay with a money order in person, but I'd like to have another option that also doesn't require a bank account. 

You should be able to get maybe 3 properties out of that, and the cash flow from them should start replenishing that savings.Aggressive house-hackingHouse hacking will be a quicker path but requires you live with others.

Lenders will most likely require all these docs.

And also proof that tenants were not harassed when they left is a requirement to the conversion.

My position would be to only allow the new resident if they meet your requirement but NEVER let the one moving out off of the lease until all of them are gone.