How would you make something like this? The flaps need to be opened. I haven't a clue how to start and not making the flaps just fall completely off.

Are you planning for them to be opened and falling to the outside or just open and standing straight up? 

You could make them out of either gumpaste or a gumpaste/fondant mixture and let it set up for days before you needed it.  I think either way they would be VERY fragile (not to mention if they are to fall open, you've got the "L" shape to deal with while drying) and depending on how/where it's delivered it could be tricky.

I might be inclined to use foamboard and cover it with fondant.  At least you would then have some strength to it and thickness.  Obviously, they would not be edible..............  I'd like to see a picture of it when you are done.

Well I didn't hear back from her so maybe she found someone else. :) I was kinda looking forward to it!
 

So I had the same question and was unable to find help... and I finally came up with this method. I've actually made a photo tutorial but I'm not sure how to upload the pictures.

I made the "box" using fondant panels with some CMC to make them a little firmer. Then I made the four flaps out of gum paste. I also made little triangles (the width of which will determine how raised your flaps will be) to use as brackets for support, under the flaps. I let everything dry hard. Then I put royal icing on the brackets and placed them at the  top edge of the box and let them dry in place. Then I put royal icing on the top edge of the box and the brackets and placed the flap on them. When they dried hard the whole structure was surprisingly strong. No transport problem.

The only glitch I encountered was one of the walls pulling away from the cake from the weight. As I had left an inch or so free at the top of the box so I could fill it, the top part of the fondant panels were a little less sturdy. I simply reinforced the  back of the panel with royal icing and made the bracket thinner so there would be less pull outwards.

Hope that was helpful and will be helpful to anyone looking for a way to make flaps!

Happy baking to all and if someone can tell me how to upload pictures I'll add them.