I have a 2000 4Runner that I installed tundra front brakes on. I also had to replace the transmission because it looks like they had a pink milkshake case two years before I bought it. When it's cold outside (below 40 or 30F) and I start the 4runner from cold I'm running into an issue where I will press on the accelerator and my truck won't budge, but then once I continue to add more pressure it "releases" and my car lurches forward with a big "thunk" sound. It's pretty jarring and something is clearly stuck.

The weird thing is, when it's slightly warmer, nothing happens. The truck will smoothly roll from a parked position to driving. If I put in in reverse, whether hot or cold, the truck will move fine. For some reason, forward driving from a cold start during cold weather will produce this issue.

I think your rear brake shoes are sticking. Time for a rear brake inspection.

Agreed. I have sticky rear brakes from a leaking rear axle and it's just like OP described.