When in doubt, ask for references or provide references (actual photos of existing 'mascot' logos, think Michelin, Jaguar, WWF, etc), to help yourself and the client identify the type of work needed, then figure out if that's something you can actually do.
This is the typical case of a vague briefing that only reveals itself in full after the terms (...
The answer is that it will either require an illustrator (capable of performing a limited number of poses and familiar with how 2d animations are executed) or a 3d artist that can create a model (also familiar with how 3d animations are executed.
For character animations, I would first inquire with the client to determine just what sort of form these ...
This is already voted as opinion-based, so here's what should be done in my opinion:
ditch the purple
recreate the icon so it has one half made of fills, and the other half, of strokes
Unless you want this converted to a vector format (which is probably not that easily doable with a single step, like an image trace in AI), you'll probably do this quicker in Photoshop, but keep it in PNG format - even better if the PNG is transparent.
Just open the PNG in Photoshop, select around the text, hit Ctrl+U, tick Colorize, then play with the ...
The only way to do this in Illustrator would be to use a gradient on path outline (stroke), instead of shape. (Post CS6)
You will have to recreate your path as a line instead of a shape.
Draw a path in the shape you want then apply a gradient to the stroke.
In the Gradient Panel Window→Gradient (Ctrl+F9), you want to switch it to Apply gradient along ...