... and, by the way, your question should be 'Is it correct?'. 'Correct' is an adjective, 'correctly' is an adverb. With the verb 'to be', we use adjectives, not adverbs. So you can say, for example: A) Is this sentence correct? or B) Have I written this sentence correctly? In A, the adjective 'correct' refers to the sentence. In B, the adverb 'correctly' refers to how you have written it.

It should be 'I will go on vacation next month'. (US English).

You can say "I'm going on vacation next month" if it is already organized. "Will" is for things that come into your mind at the time of speaking.

I will go on vacation next month---correct I will go in vacation next month.--incorrect