It is possible to do this in AE.... PROVIDED you have the right kind of card. Otherwise, Render times skyrocket. Even with a card, it takes ages to render.
You should find your answers here.
Look for the section titled, "Vector art footage to shape conversion"
If you find that it's all Greek to you, you're not alone: as a newbie, you should be made aware that there are very few things in AE that work intuitively. You may want to pay a visit here to get a good AE foundation.
Aside from all the technicallities, a better question is what and how you want to animate all the sub-elements. Sure, you could extrude it as one big blob, but that wouldn't be particulalrly interesting. And I hope you are awre of the typographic deficiencies of your artwork like the "o"s being to fat and the A and the I melting into one another...
Thanks for the critique. I am aware of those things you pointed out. I did not design the logo, but sometimes (this time being one of those instances) things like the A melting into the I are done on purpose. I attribute the "o" to the font that was used.