Redrawing yes, but it's mostly type on curves, and a few standard shapes (circles, star). If you have the typeface, it should be quick and easy. If not, maybe find out what face it is exactly and buy it. Then you get to learn about type on a path. :+)
Not sure if you still need the help, but here is what you should do.
Open the image in Photoshop. Make it a complete black and white. Enlarge the image - > save an alpha version. Basically you want 2 images, Black & White and the other Black & White, but inverted. Then, bring in the new large black and whtie into illustrator by placing it. Select Image Trace and choose any of them that yields the best result. You will then need to do the same for the other image to get a cleaner shape. Put them together correctly and you got your graphic. You may succeed with one image sometimes.
You will still need to run pathfinder and clean up your paths. Adjust the black and white into color and done.
Credibility - tradeshow design/printing involves fixing a lot of bad logos from various clients and customers who do not have vector versions for scaling.
I have attached a EPS version of it for you in black and white. =)