I want to eat alone while on vacation to have time away from my family
Are you okay? Why do you want to eat alone?
I just need time away from my family because they're bothering me.
If you feel like you need to take some time for yourself during vacation then its okay. Just be honest about your reasons for doing so when telling your friends or family. Try making it up to them by planning some events with them during your vacation.
If I do that, will they let me?
It depends on how they are feeling, maybe they'd want the time off too to be by themselves! Wouldn't it be nice for each of you to have your separate activity, and then tell each other the great stories of what happened?