Inspired by this question and by the novelization of The Force Awakens:
The novelization of The Force Awakens plays up the problems Finn has in understanding Chewbacca's native language of Shyriiwook. Chewie is clearly incapable of speaking English/Basic, but I see no reason why a literate Wookiee who understands English/Basic wouldn't be able to write out what he is trying to say, at least in some situations.
Almost any technology depicted in the Star Wars universe tends to seem "analogic" instead of "digital".
However, when you repair and work with technical and electronical devices, you need to understand tools readings, just to see if the correct voltages apply to a circuit, or if the temperature of oil is correct, for example.
This suposes already a certain level of literacy, at least being capable of understanding numbers and symbols.
But I think that the key is astronavigation. We don't see exactly what they do in the ship to jump to hyperspace, but it seems from novelizations that several complex calculations have to be made. Surely navigation computers make the hard math, but the pilot need to feed the computer with data info about where are we and where we want to go. This implies to be able to understand a estelar cartography and comprehend at least basic rudiments of estelar navigation. I don't imagine that an illiterate can achieve those tasks.
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.
