For example, I will go Mexico for vacation next month. Vs. I will go on vacation to Mexico next month. Is there any difference? If not, which one is more common In colloquial AmE?
To go on vacation is to take a vacation.
But in my AmE dialect questions like these:
Where are you going for vacation?
-- I'm going to summer camp.
What are you doing for vacation?
--I'm working as an intern in a biology lab.
would be used when the vacation is not at an arbitrary time during the year (as the person taking the vacation sees fit) but at a time that occurs regularly, such as "summer vacation" (the summer months when school is not in session) or "winter break" (the time between Fall and Spring sessions).
What are you doing for winter break?
means "What are you planning to do during winter break?"
In this pattern, for is complemented by a noun-phrase which refers to a time-span.
What are you doing for the year?
What are you doing for the summer?
What are you doing for the Fourth of July?
What are you doing for the next hour?
What are you doing for the next fifteen minutes?
In my experience, "go on" vacation is more common. However, both will be understood:
"I will go to Mexico for vacation" means the same as "I will go on vacation to Mexico."
As a side note, it is even more common to say, "I'm going on vacation to Mexico next month," replacing "I will go" with "I'm going."