on the summer vacation is impossible in that isolated sentence!
You would need a special reason to change it. For example, if you had
already spoken about
two vacations - a winter vacation and a summer vacation - you might say:
I was very lonely on the winter vacation, but I made some friends on the summer vacation.
This is a rather unusual usage, however. That is, it's not often
that such a situation would occur, giving rise to the need for "the"
within the idiom "on vacation".
CJ