There are way to many minor things that can cause a bad experience and when someone wants to sell their share or cash out without much notice it can be tough.If you can borrow the money and plan to pay it off/back by 12/24 months it would be best versus co-owning. 

Quote from @Cynthia Lee: Accepting even $1 could be seen as evidence of a payment plan, and the entire eviction process could be derailed.Once you start the eviction process, to not accept any payment unless it is a full payment in certified funds (cash, money order, cashiers check).

Keep in mind that landlords in expensive markets aren't making an profit (cash-flow) on rent.

You won't be able to use conventional financing and must use a non-recourse loan which typically require 40% down.

For the experienced cohosts in the group, I’d love to get your thoughts on payment processing options for cohosting companies (LLCs) that position themselves as property management companies but operate without a broker’s license.As an investor and soon-to-be STR entrepreneur, my business will initially focus on managing a small portfolio of vacation rentals with plans to scale. 

I recommend an electronic payment method.

Or would you not try to do this deal at all and wait to build up the cash position to fund the down payment, which could take 2-3 years?

Non-profit or not, it will be UW as a business...the non-profit designation doesn't eliminate the lending review, just makes the proceeds non-taxable in order to be put to charitable use.

Are there better returns to be had renting to EDEN housing assistance renters, or should I stick to cash-paying tenants?

To fund additional investments, consider side income, creative financing or creating a timeline with your co-owner for converting property B into a rental to generate cash flow.