Far as I am concerned there is no question here. I would give her the money in a heartbeat. It's not like she is needing the money to go on vacation with her friends, she needs it for her education. Education trumps fun every time, besides family vacations will come and go, an opportunity like this for your daughter doesn't.

Not sure how old your other children are but if you explain it right they should understand. Maybe you can do something special once a week for a month just to make up for them not being able to go on vacation.

Congrats to you for having a child that wants to learn!

Yes.

While a family vacation may seem equally as valuable, there are few educational opportunities as valuable as traveling. By taking a class in Europe, you will be exposing your daughter to a much larger classroom and significantly extending her education.

Here is what I would do:

Give her the $4k now and continue to save.

Save enough money so that you and your wife can travel to Europe yourselves for a week at the end of her month.

This way you won't interfere with her travel / school but you'll still be able to enjoy a little bit of vacation time with her.

She can act as your tour guide. She'll know the best cheap hotel, know what to see and do, and will be able to convey much of what she has learned.

In your shoes, I would have to ask myself the following questions before I could answer the big question about giving her the money for the class in Europe:
1) Am I prepared to provide the same opportunity or equivalent funding or other opportunities to my other two children when they reach an equivalent age? What I do for one, I need to be willing and able to do for all three.

2) If you cannot afford the same opportunity for all three children (at appropriate times), perhaps your daughter could pay you back some or all of the amount over time (after graduation, perhaps) to help fund opportunities for her siblings?

3) How will giving up a family vacation impact each of you individually and all of you collectively? If you can postpone for a year, perhaps there would be no harm?

There is no doubt that this is an extraordinary opportunity for your daughter. The only question becomes whether you can maintain fairness and equality between all your children without showing favoritism. If so, then wish your daughter bon voyage and tell her to take pictures and write in a journal so she can help all of you experience this opportunity vicariously!
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Perhaps since the decision will impact the family, the family should be involved in the decision, even though technically money management is the parents responsibility.

Ask the other children how they feel about losing their vacation time and allowing their sister to go to Europe for a month. Perhaps they will think it's a great idea and this way when vacation time rolls around, they won't be sad because they lost their fun. On the other hand, maybe they see her as the favored child somehow, and would begrudge her the money since they, in turn, will get nothing out of it. If they do not agree, perhaps some other compromise can be reached, such as matching the funds your daughter is able to contribute. While studying abroad is a wonderful opportunity, I'm not sure it should be at the expense of the other two kids.

I'd say yes. However, why not consider loaning it to her interest-free instead? Perhaps she can pay it back when she's finished school. Another option would be to give her $2000 outright, then take back $2000 in an interest-free loan. You could adjust it down to $3000/$1000 or even $3500/$1500 if you think that's more fair. That way, your vacation fund doesn't have to be completely depleted, and she still feels somewhat responsible for making the trip financially feasible (instead of it being a gift).