Wedge was born in Binghamton, New York. He attended Fayetteville-Manlius High School, graduating in 1975. He received his BFA in Film from State University of New York at Purchase in Purchase, New York in 1981, and subsequently earned his MA in computer graphics and art education at the Ohio State University. He has taught animation at the School of Visual Arts in New York City. Wedge is co-founder and Vice President of Creative Development at Blue Sky Studios, one of the premier computer animation studios and producer of Wedge's films. He is the owner of WedgeWorks, a film production company founded by Wedge.

In 1982 Wedge worked for MAGI/SynthaVision, where he was a principal animator on the Disney film Tron, credited as an image choreographer. Some of his other works include Where the Wild Things Are (1983), Dinosaur Bob, George Shrinks, and Santa Calls. In 1998 he won an Academy Award for the short animated film, Bunny. He is also the voice of Scrat in the Ice Age film series, performing the character's "squeaks and squeals".

Wedge will direct the upcoming animated film The Leaf Men and the Brave Good Bugs for 20th Century Fox Animation, which is based on a William Joyce book.