Credit Union is one of the best alternative for people who have been listed in chexsystems list.
In Arizona, Desert Schools Federal Credit Union has some branch locations which use Equifax instead of ChexSystems to verify new accounts. At the branch locations which use Equifax, your credit will determine what type of account you are eligible for.
At the branch locations which use ChexSystems, you might try requesting a savings-only account instead as aparently Equifax is used for savings-only account verifications. With a savings account you can use services such as PayPal which use EFT (Electronic Funds Transfer) and ACH (Automated Clearing House) transactions. If you learn which branches use which agency, let us know so that we can post that information here.