Press RT to bring a tree life, not every tree will produce acorns but you will know which ones do as they have a yellow glow around them. There is no time limit for collecting these acorns however they will loose their glow after a while.
Don't worry about this achievement i obtained it by the second area. However here are some tips.
I noticed that killing healthy tree then restoring them has a chance to produce Acorns.
When you die bring the dead trees back to life they should net you a few acorns.
You dont need 100 all at once, I opened an Acorn door and had only 62 Acorns in my inventory when the achievement popped.