Ignoring the question of should the person do this, I would say to take a copy of all 3 CRs and your TU and EQ FICO score to a local CU and see what they say. They may give you an idea of whether they would finance you or not and it wont cost you an INQ.
If you do buy a new car, I would go for the cheapest model they offer, bare bones if you will. This will mitigate the higher payment if you can't get a good interest rate. I would look at say a Kia since its much cheaper. You can always upgrade latter when your situation improves.
You cannot ignore the question of "should" OP do this. That was, essentially, the core of OP question if you read carefully. Here are excerpts that were key:
"...we are in need of a newer vehicle. I currently have 3 cars..."
"...we have found ourselves somewhat stretched the past couple years."
"...our credit kind of sucks...we are showing 30 days for the last 5 months on our mortgage"
"...strained an already stretched budget ."
"Ideally we would have liked to postpone a car purchase until these things have been cleared up..."
"The recently passed “Cash for Clunkers” program seems like a great opportunity...."
"...unfortunately we feel under some pressure to move pretty quickly."
OP has stated that they would IDEALLY like to postpone a purchase until their credit issues and finances are fixed, but is feeling PRESSURED to act quickly due to the CFC program which SEEMS like a good opportunity.
The fact is, that if the OP pays substantially higher interest on the loan, in addition to higher insurance costs on a strained budget, that the cost of loan will be greater than the benefit of the CFC credit.
To put it another way: By pressuring the OP to take a loan they cannot afford now, at higher interest, does not provide any REAL VALUE, because the CFC credit will be eaten up in interest and loan costs AND OP will have a payment now when the budget is all ready strained.
What the various posts in this thread have sought to do is show OP ways to measure the real value of CFC now, and not act solely because the CFC is available, but to act in OP best interest, lowest costs and greatest bottom line value.
In the end, OP makes the final decision. But would hate to see OP in a loan and deal only to find out later that CFC actually COST MONEY, not providing any value due to offsets in loan costs while credit issues and financial strains exist.
Otherwise, and auto loan is pretty straight forward: Make an application at a CU, online, dealership and see who will offer you what. There's no secret there.
Good luck OP. If you do go now with a purchase/loan, be sure to get the credit fixed asap and look for a refi at better rates within 6 to 8 months so that you don't pay the full cost of higher rate loan.
So the hybrid between DO and DON'T is, get best deal that you can now while getting credit fixed, then refi asap with better rates when you get mortgage lates and credit glitches fixed.
Just trying to give the OP some ideas on how to see what they can qualify for without an INQ and a cheaper alternative to the car they mentioned.
If the OP asked me directly and I was in their position, I would not buy now.
The choice to buy is the OP's.