Can I Get Porcelain Veneers on Upper Prior to Getting Invisalign/Braces to Straighten Bottom Teeth with 3rd Degree Bite? (photo)

In your situation it is best to get the Invisalign before the veneers. I know this might cost more money up front and take more time, but in the long run you will be better off. 

If you decide to place the upper veneers first you may have to grind down the lower teeth so that the veneers to bang into your teeth when you chew.

Teeth with veneers can be moved as easily as natural teeth.  But the shape and size of the veneers may not be appropriate once the teeth are realigned so it would be wise to complete orthodontic treatment prior to cosmetic treatment.  Also be sure and be committed to retainer wear as the teeth that are veneered can shift following orthodontic treatment.

Due to the current crowding and the crossbite, it would be best to have ortho first. It is ideal to have the teeth in the the best occlusion possible, that way you can achieve better esthetic results when placing the veneers.

Since you have a cross-bite relationship between your upper and lower teeth, and since BOTH the uppers and lowers are crowded, I would advise you to have the orthodontics (braces or Invisalign) done FIRST, before doing the Veneers, Most likely, you will need both your upper and lower teeth moved around, and that should all be done BEFORE the Veneers.

Do the orthodontics on top and bottom first.  For best results, your upper teeth should ovelap your lower teeth.  Otherwise, if your bottoms ovelap the tops or meet edge to edge, the veneers are sure to fail. 

When placing veneers, it is best to have teeth already in their final position.  Sometimes veneers are called "instant orthodontics", as they make the teeth appear straight (but they didn't actually move).