"let's vacation together" this is not natural english. it's much better to say "let's go on holiday together" or "let's travel together"
Please flag your suggestion all the same.
But just to be clear,
휴가 : day off, on leave
연휴 : holidays, long break
여행 : travel, trip (mainly overseas). Commonly used to refer to holidays trips.
여행하다 - go (travel) on vacation/ holidays, vacation*
The use of Vacation is a bit ambiguous because this word could indicate either taking a trip far away from home/work during a long break; or having a stay-at-home break, having a "staycation" (스테이케이션 / '호캉스족' 휴가), that which obviously, 여행 does not cover...