There's a blog post here, which Ryan Britt suggests that most, if not all citizens in the "galaxy far far away", are in fact completely illiterate due to the available technologies, for example holograms, which store memories, thus eliminating the need for reading or writing.
However as we saw in The Force Awakens, the rebels can read maps. It's not conclusive evidence that they can read, the map was mainly illustrative, however it does prove that they are able to follow technical documents.
Further more in Episode IV, Darth Vader interrogates Leia for the Death Star "plans". If they are anything like blue prints, we'd assume they would contain some sort of technical specifications and dimensions, which requires some literacy skills if not to read, then at least to write, if of course they were written for human eyes, and not some droid language that is converted into Death Stars.
Chewie is able to not only operate, but also repair the Millenium Falcon. In our world, this would assume some sort of literacy skills, however we don't know how Chewie learnt those skills. He could've memorised them visually by learning from holograms, or by watching Han.
The answer is inconclusive, but we can potentially speculate that Chewie can at least read to some level, to be able to repair the ship. Unfortunately there's no clear cut answer, as there's very little information available that discusses anyone's level of education.