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I think your chances are very good. $20-26K in yearly salary with a $8-13K loan and no other expenses should be good. Your accounts are on the small side, but I think you've got a good chance. If anything though, I'd look for a cheaper car. You don't need a $20K car right now.
Not trying to be parental here but I should mention that when I was 16-28, all of my friends were spending crazy money on cars while I was putting money away to buy my first condo for cash.
20ish years later, I am basically retired in my 40s and my first condo I bought for cash is a long term rental now that has paid me probably over $120,000 over that time -- if not more.
If you're young, consider buying the least car you need versus the most you can afford -- and sock cash away towards never having a mortgage.
A nice car is a nice car, but it's still an anchor cost that can hold you back from earlier success than others will experience -- or that they may never experience because every salary increase just means buying more "nice" things.
If you're putting that much down, you should really not have a problem getting good financing. You loan-to-value ratio will be very low and it's a secured loan. I don't think a used car in the $15-20k range is that extravagant and over time could cost less than buying a lower cost car after you factor in repairs and the maintenance that comes at higher mileage. PenFed is a great place to start (anyone can join by making a charitable donation if you don't otherwise qualify) and their rates start as low as 2.49%
You can get a really nice car for $7k and skip the car payments and interest.
Mind you I say this having been car poor for essentially all of my 20s. I truly believe cars (and I LOVE cars) are the biggest thing people waste money on.
Additionally with your age, you will get hammered on insurance. I often had monthly insurance payments that exceeded my car payment before I turned 25. Be sure to take that into account.