Maybe things have improved since I commuted from there, but its funny that you asked because distinctly recall being drenched while walking through those lots by passing cars even more times than it had happened in Manhattan!

Yes, New Castle Residents can purchase up to two permits per household. You are not guaranteed a spot, however there have not been complaints to the Town Clerk about this. Most commuters find a spot closer to the station if they arrive between 7-7:15am. Otherwise there are lot places toward Washington St. and there should be a spot. (5 min walk). Residents can also purchase a Meter Spot for approx. $5 and then put quarters into the meter between the hours of 7am-7pm, after those hours it is free (with resident meter sticker).

Here is the link and all the info that you need for parking at the Chappaqua train station.

http://as0.mta.info/mnr/stations/station_detail.cfm?key=148