We use one word "holiday," to mean both a break from work and from school.
In the exact sentence that you have quoted "We often go abroad for our summer holiday," then you have to use "our."
However, if you were to add an "s," to holiday to make them plural then you could say:
"We often go abroad for summer holidays."
In your original sentence, you could substitute the indefinite article "a," for our:
"We often go abroad for a summer holiday."