Small update, the person who made the Crash toy was not involved in the production of the Lemmings toy... That unfortunately means pretty much all points of contact I have are exhausted.
I'll keep my ears out, but I don't have a ton of hope. If anyone finds this thread in the future and wants to contact me, my discord username is: metalgearsolid2_sonsofliberty

A small update; Thank you to doublemover (2xmvr on Discord) for tracking down a Crash Bandicoot buddy of a similar style and various other (non-Sony) toys of different products. I managed to find the email of the person who made the Crash toy originally, and given the connections I'm hopeful that he created the Lemmings toy, too. Fingers crossed he gets back to me.

Hello! I'm Leon, and I'm hoping to find a lost Lemmings application from around 2006.
This application was a 3D rendered Lemming and featured a clock, calendar, screenshot viewer, and soundboard. It could walk around, as well as be dragged around the desktop. It used to be downloaded at https://savethelemmings.com/, however the download itself was not archived. I assume it ran on Windows.

If you remember downloading it from this page, please let me know!! I would love to archive it.
Here is a video in case that jogs anyone's memory as well: