Can I take kids my children on vacation out-of-state without their mother's permission?

Why not offer to take the mother along on the trip to Disney?  Beats attorney fees.  You will need to resolve custody issues in the future unless you and the mother can come to your own agreement.  Once again, beats attorney fees.  

If you got a really good price, it's possible that you could spend more than you've paid, to have a lawyer go into court on short notice for an injunction, and I think it's unlikely that you could win this.  Your lawyer would have to (1) prepare and file a complaint for a custody or visitation ruling; (2) prepare and file an application for an Order to Show Cause, that would include an affidavit from you, and a written legal argument that the lawyer would have to research and write, and a proposed form of the Order to Show Cause.  The papers would have to be served on the mother, and the court would have to be persuaded that a sufficiently serious "irreparable injury" would result, and that would have to be based on more than just the lost money.  I'd be very surprised if you could get a lawyer to take this on without thousands of dollars up front, and it is absolutely not a do-it-yourself possibility.

If there's no order in place, unless you are married to the mother, you don't have any legal custody of the children at all, until a court orders otherwise.

Without a court order in place you are free to take your childen to Disney or just about anywhere else.  Your children's mother could try an obtain an emergency court order to stop this (an "injunction") but since this is a vacation and not a permanent move, and you are not traveling outside of the U.S., I doubt that she could get any such order; at least not without misrepresenting the facts.  This is not an emergency situation.

At this point I would just plan on taking them.  Of course, there is nothing stopping her from taking the kids off somewhere first simply to spoil your plans.

As soon as practicable, you should go to court yourself to establish some type of temporary/permanent custody.