I just traded my 2002 Crown Victoria LX Sport a few months ago. Same car as a Grand Marquis, except for some trim items. Loved it. One of the best cars I've ever owned.
There's always the obvious stuff to look for like repairs, CarFax reports, signs of rust starting, signs of neglect, etc. But a 2007 with lower miles should be fine. Since they're body-on-frame, minor
accident damage isn't as big of a deal as unibody cars.
Mechanically, they're fairly bullet proof, as long as they get proper care and feeding. See if there are service records, which would be a big plus. The Grand Marquis LS will be nicely equipped and is the equivalent of a Crown Victoria LX.
A nice option to have is the "high output" engine, aka performance package. It's basically dual exhaust and possibly a higher (numeric) rear end. The dual exhaust is easy to spot, just look at the back of the car and tail pipes on either side. If it only has a passenger side tail pipe, then it's not high performance. The difference is only about 15-20 HP, but I've driven both and the cars with dual exhaust are "crisper".
Another great option is the air springs in the back. They auto level the car. I once had 12 cases of water in my trunk and 4 people in the car. It didn't sag one bit. More importantly, it didn't handle like there was an extra 700 pounds in the car. I'm not sure if they can be retrofitted, but most likely. Even if you never load people or cargo into it, I prefer the air spring ride.
There really isn't a weak point on them. Transmissions are good for 120,000 miles or more. The rear ends are solid. The engine can handle a few revs now and then, but usually just lopes along, ready to deliver power when you need it.
In the 7 years I owned mine, I had to replace the A/C compressor. I lived in Vegas at the time, so it ran a LOT, and this wasn't a surprise. It wasn't horribly expensive, either. I had a few minor front end issues which were mostly wear items, bushings, etc. I was religious about maintenance. Mine had Goodyear Eagle RS-A tires when I got it. I stayed with them when those were worn out, and that set still at around 40% with over 40,000 miles when I traded it. Aside from wear items and the aforementioned repairs, that was it. And I drive my cars hard.
Personally, if I were to buy another Crown Vic or Grand Marquis, I'd want one with fewer than 50,000 miles. That seems to be the sweet spot.
Oh and mine got 23 MPG on the open road when my wife drove it.