No there's not
a lot of evidence against dear ole Will - you think Francis Bacon or Christopher Marlow would let an 'uneducated bumpkin' take the credit for so many works -
A few maybe, but all that body of work, which was enough to keep a company of jobbing actors going commercially for so many years? Don't think so - the Elizabethan Age was on a par with Gordon Gecko's Wall St. in terms of people on the make, so no way would he have got away with that big a con at the time.
As for the general lack in education for 'common people' blame the Roman Pope and the religious wars during the Renaissance - in medieval times even the very rich could not read, male or female - that was the province of priests though some lords did learn to read and write more than their own names if they showed an interest. Maybe the regional overlords and royalty did make the effort, but there were no schools as such and those that did teach were mostly based around the abbeys, many of which produced handwritten books and manuscripts which weren't all centred around the Bible but did not stray far from morally correct subjects (note that political
correctness comes along only when Kings started to want to run the church in their own countries
Only the truly rich could afford books because of this. The first printing presses changed this forever and as feudalism died out learning to read and write became more widespread - part of the reason for that was because Protestants were on the move and part of their beef was that they wanted the Bible to belong to everyone and as that movement found a lot of favour within the merchant classes the demand to read and write and decide whether the gospels were the books for them meant that these skills were sound business practice if nothing else. Some of the more enlightened estates had their tenants being taught by scribes, some of them former monks. So an improvement in educational opps around the time of Shakespeare's birth.
As for the man himself - well he certainly wasn't dirt poor... Mary Arden's family were landowning farmers and Anne his wife was from tenant farmer stock (so his family were 'richer' than hers and pretty respectable), so they were both relatively well up the social scale and it's entirely possible that William was taught to read and write pretty young. The fact that they married at all (Anne was definitely in the family way) is an indication of status, since the working classes rarely bothered to get wed by licence which William and Anne did, presumably due to family pressure
So not so much poor kid done good as he probably had a goodish start in life for those times