if you have access to after effects, try doing it there - a really quick job

if i had to do it in maya,
i would export illustrator file as a hi res .png image (4k) and assign it to a shader in maya (surface shader). Assign shader to a poly plane, make a basic animation and render it out.

but really for a task like that one - since you use illustrator i assume u use Photoshop.
try looking up a tutorial on youtube animating in photoshop. It will get you what you need. Photoshop can do basic animations, it can do that

here, there is a simmilar topic
https://simplymaya.com/forum/showthread.php?t=36947