Sentence 1 is not correct, for the reason given by Mitsuko. "While" needs a clause, whether finite (with a verb marked for tense) or nonfinite (without a tense-marked verb) Sentences 2, 3, and 4 are all correct. "While" can certainly be used with a prepositional phrase such as "on vacation" denoting a state. It is also used with adjectives and participles:
While sick with chicken pox, Frederick managed to support his family by doing some freelance editing
While flying to her next reporting assignment, Susan realized that she really wanted to retire to a desert island
Sentence 3 can be considered a reduced version of 4.
The "reduced" adverbial clause is common in written style, but not common in casual conversation. That 's why, for stylistic reasons, the full clause, as in 4, would be preferable in informal style.
What is not correct about all the sentences is the use of "the" in "on the vacation." The correct expression is "(be) on vacation." One is on vacation, on leave, on trial, on duty, etc.
(It is also possible to use a possessive:
They visited Paris while on their vacation
The possessive is unnecessary, however, since the state of being on vacation is automatically assigned to the grammatical subject of "be." You can't be on someone else's vacation.)