Since when "vacation" is a verb?
English is an action language; many, many nouns can be used as verbs.
You speak about american english, don.' t you?
While that might be true, I've literally never heard someone use "vacation" as a verb and I would think people will be slightly confused once you do.
The term "action language" is actually a thing? As in used in linguistics?
Why cant we say "함깨 여행합시다" ? Does it mean let's go vacation together?
You can, though Duolingo here is looking for the sentence at the Causal speech level.
Would 이랑 work or must that be used in conjunction with a noun?
(이)랑 requires a noun. It is essentially the intimate version of 하고 and 와/과.
여행을 같이 가자 this should be correct right?