is there a name for the under eye implants/aegyo sal procedure? (Photos)

This is not something that is traditionally desired in Western cultures, but, it is not hard to create. Filler injections should be able to do that for you if placed by an expert. Or, fat grafting or even fat repositioning may be able to get those results.
Should be an office procedure.

You are referring to what is commonly known as love band surgery. This is performed by placing Alloderm grafts under the skin but above the orbicularis muscle of the lower eyelid. While it is commonly requested by Asian patients due to the cultural perception that such a eyelid roll is youthful, it can be performed on anyone of so desired.