Both are correct. The difference is when you use an article, you are speaking about a specific vacation.
So to say "go on vacation" you are refer in general to a vacation, but if you say "go on a vacation" you are implying a specific vacation. The difference is subtle, but perhaps:
"I have been working too much I need to go on vacation." -- you are not planning it yet, you just need one
"This summer, my family and I are going on a vacation." -- you are actively planning the trip
Again, very slight difference.