Is there anyway to get rid of or reduce the periorbital fat around both my eyes without surgery? (Photo)

Thank you for your question.  Unfortunately surgery appears to be your best option.  There are no safe and effective methods to reduce fat around the eyes.  Best of luck.

Eyelid puffiness can be corrected using a surgical approach. Most plastic surgeons will prefer this option because it provides a longterm solution to your problem. Best, Dr. Karamanoukian Los Angeles

You look like a thyroid patient, do seek opinion about this from your GP or endocrinologist. Surgery is the only option.