While the company must abide by certain federal laws, including the Fair Credit Reporting Act, it has not been regularly supervised or subject to on-site audits by any government agency. But the Consumer Financial Protection Bureau, the watchdog agency that Congress set up two years ago to protect consumers from harmful financial products and companies, hopes to change this.

The CFPB announced a rule last month that will allow the agency to supervise consumer reporting agencies, including "specialty" ones like ChexSystems. Under the rule, the consumer watchdog will be able to conduct examinations, investigate any unfair business practices and write new rules.

"Federal supervision will not only be important, it is a game changer," said Wu.

About 7.7% of U.S. households -- or 9 million -- are unbanked, meaning they don't have checking or savings accounts, according to the FDIC. And Wu said many consumers denied accounts are becoming part of this group.

"The people who are denied fall off the mainstream credit and banking map and join the ranks of the unbanked," said Wu.