That would be 으로, not 오르
오르다 does mean "to climb", but in a more general sense. Like climbing a ladder, stairs, ... Could also be a mountain but there is a more fitted word specifically for this purpose.
등산 means "hike" or "mountain climbing", then you add 하다 which is "to do" -> 등산하다, literally "to do hiking" or "to do mountain climbing". Then you conjugate it to the propositive: 등산하다 -> 등산하 -> 등산합시다
In common use nationally in Korwa this verb means "to hike", not rock climbing or Mountaineering. Most Americans use the word "climb" to mean rock climbing or mountaineering, so the better translation for Americans would be "hike" only. I struggle to correct this when I was teaching Koreans for many years, but they still use the word "climb" when clearly it should be "hike." My guess is that the program is double translated from English to Korean and then Korean back to English, so we have some of these leftover bad translations that Koreans use.
Since 등산하다 is a noun-verb, 등산 means mountain climbing. So, 등산하다 literally means "to do mountain climbing".
In English, we can say "Let's do (or go) mountain climbing (or mountaineering)".
So, imo, "Let's do mountain climbing together" should be accepted for this question. Idk if it's already accepted but I'm too doubtful to type that in. Even the hints offer no clues.