I would never put 37s on with stock front axles even with an open differential if I planned for it to be more than a mall crawler.  Having said that, I chose to not run a full locker until I upgraded my front axles.

I can tell you that my stock axles survived quite a bit of wheeling with 33s and a limited slip front differential, but I'm as kind to my truck as possibly while still taking it where I want to go.  I've seen 37x14.50 Toyo MTs take out one stock and two chromoly front axles in Moab in two days.  Once the front axles were upgraded (overnight delivery) the front ring gear  took some damage.  I honestly can't say that the guy driving was being much if any harder on his 4Runner than SqWADoosh is.  More aggressive than I choose to be, but not certainly not thrashing their rigs.

SqWADoosh is absolutely correct about RCV front axles being the best.  I am running stock outputs with my lockers and 35s so can't give you any opinion on the need for upgrading your outputs though if you can afford it stronger is pretty much always better!

My motto is build strong, wheel easy, and hopefully avoid broken parts.