Well. This question made me sad.
One of the best local theaters to get involved in is now no longer: Manhattan Theatre Source. Unfortunately, the independent theater scene in New York has taken a mighty blow over the last couple of years due to a lack of financial resources.
Some other potential places that I know of (but I’m not 100% sure if they have an open-door policy as MTS did) are Axis and The Living Theatre and maybe even Naked Angels (although, while this is a great place to see readings of new playwrights work it can be a little cliquey and I’m not sure how open they are to new people – I haven’t been involved with them for many years though, so things may have changed).
I’d say the best resource for current theaters that you can get involved with and audition for is Backstage. The hard copy might be easier to look through than the online version. You can find it at Drama Books or wherever newspapers are sold. New editions come out every Weds night or Thurs morning.