I work in the industry. The short answer to your question is no, to the best of my knowledge no such card exists. As someone else noted, debit cards can sometimes work this way, but not credit cards. As a rule, rewards cards are intended for people with good credit and significant spend. The underlying economic models are completely different for a subprime customer who tends to revolve or one that presents some risk and those which are essentially zero-risk and never carry balances. Rewards are quite expensive to dish out.

My advice would be to get a secured card for as much as you can swing, stick with it for 6 months (use it, but pay it off in full each month) and then call the issuer requesting an upgrade to a better product. Six months is the bare minimum here really - YMMV - 12 might be better.