You don't need a petition. You can just submit a proposal to Unicode.

I did it for https://unicodepowersymbol.com/

Seriously, these petition sites are just data harvesters.

Not to be a jerk, but the image used there is a pretty bad example of the Cool S. It shouldn't have horizontal ends, and should instead use diagonal lines to connect back to the middle.

Not that there's exactly one Cool S, but this does not look like the most common or recognizable variant.

Origins unknown, the cool S is to be tribally handed down through middle school hallways. This is the way.