What is the Best Non-surgical Treatment for Sagging Underarms?

Thank you for your question. There are a number of radiofrequency and infrared devices that use heat to tighten the deeper dermis. This can create some mild skin contraction and tightening. These options usually require multiple treatments to achieve a satisfactory result and then maintenance treatments. And it also depends on how much tightening you need. You may not be a candidate for this at all. 
Liposuction alone removes fat and if you're young and have good skin tone it may contract. 
Most patients are best served by a brachioplasty or arm lift. This can come in a mini version if the defect is minor. The trade off is the scar but the outcomes are more dramatic. 
See a board certified plastic surgeon for consultation to learn what might be best for you. 

A brachioplasty is probably what you are looking for to achieve significant results. There are really no effective non-surgical treatments for removing significant skin under the arm. Non-invasive procedures like coolsculpting only address the fat that lies under the skin and over the muscle, it does not do anything to improve excess skin. Laser therapy and other skin tightening procedures show extremely minimal results, if there are any results at all. There is a reason why board certified plastic surgeons do not include this procedure in their practice offerings. It would be best to consult with a board certified plastic surgeon to review the best options for you.

There are several treatment options out there for sagging underarms but do not involve surgery. The  goal with these treatments is to maximize the results with the minimum of downtime, pain or disability. Unfortunately, as with most things in life the reality is… There is no such thing as a free lunch!
The treatment options can provide some modest improvement in skin sagging and may be used in patients with a minimum of loose skin, but rarely provide dramatic results. Common treatments used are radiofrequency, laser and possibly some injectable agents such as Kybella.
 It goes without saying that the most definitive results come from arm lift surgery. Granted, there is downtime associated with the procedure, for a complete and customize result that is generally the best option.
 As with any surgical or nonsurgical procedure, it's critical that you seek the advice of the surgeon who is certified by the American Board of plastic surgery and has extensive experience in body contouring.
 I wish you the best,
 Dr.  Gabbay.