You can absolutely get a loan, but you're likely going to need to put around 20% down, and you may still be stuck with a high APR loan even then.
Used car financing is problematic, I mostly found that a dealer was probably the best way to go, but I still got stuck with 19% as a rate, with plans to refinance in sixish months off that number. Most auto-lenders can be subprime friendly with enough down, but some want more like 50% down (Chase), Wells was fine with what I did (35%), and Roadloans gave me a loan which worked out to 33% down. I don't know what the Chase rate was but the others were right around 19-20%.
Edit; admittedly I was shopping for a more expensive car than I truly "needed" as far as used cars go, YMMV on downpayment if you aren't looking for a ~30k car.
I also think lendingtree isn't a bad solution, as it'll hit Cap One and Roadloans (two traditional subprime lenders), but you're stuck going to dealers of their choice on approval, and that's not always going to get you the car you want. If you're a member of a credit union or bank, it's not a bad idea to check with them too possibly.
The most salient advice I can give, is don't pull the trigger until you're ready to actually go: regardless of how many inquiries you spam out, they'll get rolled up from a scoring perspective within a two week period (maybe longer but I don't bet on it). That said, everything can be done in 2-3 days easy on a car loan, assuming you have a downpayment. Cash up front is the route to approval when it comes to subprime used auto loans.
See if you can get a "regular" loan through your CU. I am also a public servant and my credit union seemed to be a bit more forgiving about my tarnished CR. I got a 7.99% APR with scores in the mid/low 600's. I had very very high Utl 1 unpaid CA and my CU does not use Fico but a Fako (of which I'm not sure what the score was exactly). Your job stability and income will work in your favor. I don't think there is a need to shop sub-prime just yet 
