Astro van won't shift into gear

Maybe something wrong with neutral switch? Usually it is on or close to transaxle shift lever, follow the shift cable to shift lever, the switch has multiple wire connector.
It may be possible to start vehicle by jumping the load side of starter relay, key on. There are safety issues trying this, the vehicle will crank in any gear. Make sure tranny is out of gear, parking brake is set, block the wheels, whatever it takes to be safe, safety is your responsibility.
If the problem isn't that neutral switch, then I don't know? Any applicable trouble codes?

http://www.motortrend.com/cars/2005/chrysler/300/recalls/
I do not see your problem listed as a recall.

it sounds like a shift lock solinoid ....this prevents the car from being shifted out of park when ur foot is not on the brake....u will have to bring this to a repair shop since the faulty part is located inside the paneling where ur shifter handle is

you have to take off the plastic all around the gear shifter. You can wedge a screw driver carefully around the edges and pry the plastic off then there is a screw on the shifter take that off and slide the top off and then take off the rubber until the whole shifter is exposed then you will see in the cavity on the right side a little metal round tab on a spring that you have to push in with a screw driver and it will release the shifter. Good luck

I had the same problem. The interlock was not the problem. I had the shifter replaced (at my own expense because drive train warranty would not cover). After examining the old shifter discovered source of problem is a plastic cam on the front of ths shifter. A tiny PLASTIC ear which connects to a small spring had broken off. I repaired this by putting in a small screw where the ear had been. Incidentally, my tow truck driver had also seen the same problem on several Chrysler 300s. In my estimation this is a poor design on Chrysler's part.