I applied at Walmart?

"I applied at Walmart?" is your lead line. With a question mark. Are you asking if you applied at Walmart. Or are you saying you did.

Obviously you did and got the job.

We would have no way of knowing if they will give you this time off or not. Being a new employee and part time - they may. They may not. Using family reunion as a reason does not mean they will care and automatically grant it.

Asking for a week off after a couple months of employment is never a good idea. Asking for that week probably isn't a good idea either - as they will be getting rid of Christmas stuff, doing Christmas returns and getting ready for the next big thing and will likely need all hands on deck.

Are you sure this is a permanent job? Or a season - through Christmas season - job?

You can certainly ask. If denied - then you can decide if having the job is more important than a trip. Hopefully, girlfriend would understand that. Sometimes plans made long ago need to be put on hold when reality pops up.

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Unpaid leave is not considered personal days or vacation. Personal days and vacation are types of paid time off.

As far as if you'll be allowed to take the time off, that's up to your managers. Usually with time off requests, the earlier you ask the better, so I would speak to a manager ASAP.

Ask them asap. Since it's after the holiday season, they might say OK