What do her psychiatrist and therapist say?

If she was showing signs of a different life threatening illness (mental illness kills more teens than cancer) that made vacation challenging would you just go without her? Force her to do something painful? Punish her? Or would you cancel the vacation to help your child?

The depression and family trip are two different issues. Sister gets to miss it so why shouldn't she is her logic. For the depression she needs a mental health screen and possibly therapy but these are two separate issues. She is 14 and she goes.

I guess you've already received a lot of replies- I wouldn't let your DD stay home at fourteen. It's too young and older sis will be at work, so little in the way of supervision.
I agree with the pps that this should be approached respectfully but w/o the impression that you might back down.

"Larla, we are going because we aren't leaving you home for four days. Let's put our heads together on how we can make this trip more pleasant for you."

Cue anger, refusal, door slamming, etc.

Also, OP, I would be reluctant to give my DC much unsupervised time if I knew he was depressed. What I wouldn't do is expect full participation- so, on this trip, I would be super gentle about my expectations, but I wouldn't leave her at home.

Let us know what you did OP.

OP with an open minded sense of level of her depression I’d present, as I’ve done to mine that she is a child and we are her parents. With that, we as a family have obligations and some of those obligations are to preserve family ties. As she is still reaping the benefits of a child in the family she has to come with you. At 14 I’d say to this child-you are on the cusp of bri g an adult and while I honor your opinions and recognize that this is something you feel strongly about, what I won’t do is force you to attend but I’d have to evaluate first whether I’d let her stay ‘with her sister’ (how many hours is she working? and what an imposition on this adult sister who presumably followed family expectations and will now be dragged down with burden of policing her sister.)

I may have to cancel trip or just send others but the consequences would be unpleasant. And if depression waa one of her reasons-than she would be at a therapist’s office for a prescribed amount of time (say 6 sessions.)

What I would not allow is her to go on trip and be such a *hit that trip was ruined.

Agree with above to not pick battles you can't win, or especially to escalate fights. Usually I give choices, just like I did when they were toddlers. Something like:

You are 14 and can't stay home alone but here are some things we can do:
1) Shorten the trip (my teens don't really mind long car rides and in this case, I'd be willing to shorten the trip).
2) Change up the sleeping arrangements
3) If you insist on staying home (and since I can't physically put you in the car), I'll be really sad to miss seeing my family and will be considerably less likely to accommodate your request for transportation or electronics over the next few weeks/months.

Sometimes, just the acknowledgment that we can't force my teens to do anything, opens up room for compromise/discussion of the problem.