No credit card required
Browse credit cards from a variety of issuers to see if there's a better card for you.
I recently got a $10K loan with an $1100 monthly income and $3K down through Ford Motor Credit. I pay $500 rent. Had a 780s-790s FICO score with 3 years of history with a single credit card. APR is a bit high at 9% but at least I got the loan. Hope this helps.
Welcome to the board. Since you are going to be putting down $5,000, the car financed will likely be $3,000 or $5,000. That is usually too low for most companies to finance. Your best option is likely going to be getting in with a credit union and applying for a personal loan with favorable interest rate to get the car you want.
I agree 100% with those above on going the credit union route, if you can't get into NFCU you certainly get into Penfed or DCU which are excellent for car loans. If you have 5k to put down I think you can get this done pretty easy. I would not go to a dealer without a preapproval in hand because your the kind of person they will shove into an artifically high interest loan (they mark up the APR to make more profit). Particuarly because you have low income be cautious in your selection of cars because you cannot afford someone elses headache. There are a lot of lease return cars on the market right now so you may be able to get a 3 year old Corolla or other economy car in your budget that would be pretty solid.