There is no grammatical reason that you cannot use "mostly" to modify "prefer" since prefer is a verb and "mostly" is an adverb. However, somehow, it sounds more natural for me to say: Most people prefer to take their holidays in (the) summer. (Most Americans would use "the" with seasons, whereas I understand that the British do not)

Yes, fine. I would usually use "usually," and put it before the verb, but your sentence is fine. So - "People mostly prefer..." or "People usually prefer..." Those sound a little better to me.

It is possible in conversation as rarely people will correct you, but in writing, it is correct and better to say: "People much prefer to take their holidays in summer."