Lets say I am wild shaped into a large snake with constrict and the grab ability, could I initiate grapple (without the bite attack/grab ability assuming I have improved grapple feat) and then deal constrict damage?
You automatically constrict, and you can bite if that's what you want to do (instead of moving or pinning). No, you don't somehow re-grab them if you bite, all grab does at this point is give you a +4 on your checks to maintain the grapple.