I have spoken to several local banks and credit unions and none have any properties where they are in default.

You set your screening standards at a level where they will most likely not qualify, as a example setting a high credit score requirement will eliminate many on disability.You keep taking applications until you find the highest qualified applicant.

What documents do they require, what credit scores do they allow, how do they verify previous rental history, etc.?

Using a service to report rent debts to Credit Bureaus can be very effective at recovering those debts because it can impact their credit report and score for years to come, typically 6-7 years depending on location. 

I would connect with a few lenders on here, find one local mortgage broker, and then call around to several regional banks and credit unions.

Until now, I had a great credit score, but I'm concerned that my increased debt might affect my ability to obtain a HELOC or a Home Equity Loan.A lender approached me, claiming they could help me secure a HELOC, but the encounter felt coincidental, making me feel vulnerable.

Prepare your finances, including a strong credit score and a down payment.

I require my tenants to have a minimum credit score of 680. 

Try local banks and Credit Unions, they'll offer the best terms for you. 

Avail.co also can do the tenant background, eviction and credit score checks.