Can I have surgery for shoulder instability on both shoulders at same time or back to back? Inherent ligament laxity. Very active lifestyle.

Back to back. Theoretically - yes, but practically -unlikely. Chances are you will need at least one of your arms/hands to be available for taking care of yourself and your basic needs after surgery. It is difficult to go through rehabilitation when both shoulders are hurting. The good news is that your second time surgery and rehab will be a much easier experience since you will know exactly what to expect.
Not advised. Unless you have someone that really loves you and is willing to wipe your behind i would not advise this and most shoulder surgeons i believe would agree. It would be strongly ill advised. I would stage this but first off make sure if truely mdi (multidirectional instability) that you have failed a good round of therapy first as many times this can improve this problem without surgery.
Let me explain. You could have the surgery on both shoulders in the same time but i will not advise it, because will not be to function at all you complete assistance. If you have a complications (infection) in one shoulder may spread to the other.