No credit card required
Browse credit cards from a variety of issuers to see if there's a better card for you.
Given your situation, its going to be tough. My suggestion is to work on your credit. Go open up a secured credit card, maybe even two. I suggest Cap one secured. That should help your credit.
As far as getting approved, I don't know, maybe carmax? But they will rob you blind on interest rate..18% or more. Maybe a used car dealership with in house financing, but they will rob you too.
Work on building credit, if you can hold off buying then hold off.
^^Agree with the above. Also, $15k is too much car for you to buy right now. You need to walk before you can run. Spending up to 80% of your income on a car payment, even if allowed to by the lender, is a bad, bad idea for you. At that payment rate, you can only end up in disaster. No one can sustain montly payment of 80% of their income for 5 to 6 years.
You have two very serious issues 1) you haven't established good credit and 2) you don't have the income to support such a large purchase. And most importantly, you have a child, which should come first. A large vehicle payment will hinder your ability to get on your feet financially.
I am not giving you a hard time. Part of being an adult is looking at the facts and determining the best course of action for you. Since you don't have enough income to support a purchase of $15k, AFTER you have opened up the secured card and shown payment responsibility. Then look for a much less expensive vehicle, well under $10k. When you increase your income and get on your feet financially, then consider getting a vehicle in the $15k range.
CarMax if you have one near by.
Roadloans otherwise you can get an instant offer with 3 options to see just how bad the financing would be online.
I would look at doing the pre-qual at capitalone.com for a credit card and see if you can pick up at least a secured card and wait a month or so.
How much medical collections are we talking... if you could pay them, you could get them deleted then they woudln't be an issue at all.