Does anyone know if it's permitted to camp at the trailheads in the Adirondacks? Basically we are looking to drive up on Friday night, park at a trailhead for a high peak, then stash the camping gear in the car and start the hike early in the morning. Can we just walk 150 ft into the woods from the parking lot and pitch a tent?

In general, yes.
http://www.dec.ny.gov/outdoor/7872.html
http://www.dec.ny.gov/regs/4081.html#13001

Check your map to make sure you're actually inside State lands at that point, not on neighboring property.
Some sites may have additional rules, check with the appropriate regional office for the DEC.
http://www.dec.ny.gov/about/558.html

For obvious reasons, camping is not allowed at the most popular high peaks trailheads. Just park near Chapel Pond on the way in and camp in one of he many posted sites there. Then you won't be far from any of the trailheads in the morning.

Get the ADK map - it has all the sites on it.