This shouldn't be a problem. ChexSystems, as its name implies, deals with checking accounts. You're looking for a secured credit card. They're two different products, and you need to find a banker with a brain.
For people who don't know, ChexSystems is a reporting service that most but not all banks, S&Ls and credit unions subscribe to. People usually get on the list by bouncing checks, but like all such services it can make mistakes.
If you have the cash to put aside for a secured card, you won't be writing checks. You'll be drawing against your own cash.
Two things: Shop around; and try smaller issuers.
Skip big credit card issuers. They are ... how do I put it ... freaking out right now. For many of them, their mortgage departments have lost billions, making shareholders rather angry. Because of their own mistakes, at this time, most are not interested in helping you. They're looking for excuses NOT to help you.
These days, credit unions and small community banks are more likely to cut you a break if you walk in with money in hand and ask for a secured card. If you have a job and can offer to do direct deposit with them, they will kiss the hem of your garment.
Credit unions are a lot easier to qualify for these days. If I were in your shoes, I'd check them out first. This is a legit credit union locator service: http://www.creditunion.coop/cu_locator/