The only cave Rabbi Lamm was referring to was the cave of Rabbi Shimon Bar Yochai (RSBY).
The cave of RSBY was a major theme with Rabbi Lamm from many decades back. I found a paper by Rabbi Lamm about RSBY and the cave from 40-50 years ago.
From a more recent writing by Rabbi Lamm, he believes everyone should spend at least one year in the “cave”. I personally disagree on this specific point.