I have to apply for holidays before the end of March
I have to put in for my holidays before the end of March
I have to put my holidays in before the end of March

Or do you mean that you have to have used up all your annual holiday entitlement before the end of March? That would then be "I have to use up my holiday before the end of March", or "I have to get all my holidays in before the end of March".

or ask for vacations

we never use "holidays" in this context - only "leave" or "annual leave" (even if it's not strictly the one big leave period every year, but only a day or two, because it's taken from our "annual leave" contingent)