There wouldn't be anything left but some very dry bones, though it might be interesting to check out what is left of his clothing. I don't there are any Shakespeare descendants, but I doubt if permission would be granted in any case.
If he was sealed in a lead coffin, common at that time, anaerobic bacteria would have liquified all of him, including bones, leaving only a smelly green goo.
When they opened up some ancient lead Walsingham family coffins to look for original Skahespeare manuscripts, that's all they found. It took days to air out the chapel to get rid of the stench.
If memory serves, his last descendant was a granddaughter who died in the 1670s. I don't think permission will be granted either. Kind of hard to ignore Shakespeare's direct wishes inscribed on his grave.