The correct option is C During
Here, 'during' is correct because it implies all through a period of time, here it is the summer vacation.
'For' is incorrect because it is used to show who is intended to have or use something or where something is intended to be put.
'Among' is incorrect because it implies surrounded by somebody/something; in the middle of somebody/something.
'Between' is incorrect because it implies in or into the space separating two or more points, objects, people, etc.