I'm assuming you mean to go enjoy and understand the play, as of course a deaf person would be perfectly able to go to the play, assuming they had the money for admission.
I can only speak to the aspect of deaf people's participation in society, as I'm not an expert in theater of the time. It is almost certain there would have been no sign language interpretation of plays, as interpreting as a profession is less than 50 years old. Although there were signed languages in Britain at the time - Samuel Pepys's diary mentions signing in November 1666 - it would not have been in wide enough use that a play might have been interpreted. If any sort of sign language was used in the theater, it would have been by a deaf person bringing along a hearing friend they could communicate with, and getting assistance that way - indeed, that's how all interpreting was done, worldwide, until the start of the profession in the 1960s.
It is possible, though, that deaf people would have been allowed to sit closer to try to read lips. This is noted by Cockayne in 2003, she found evidence that deaf people were sometimes allowed space in the first few pews of a church. This is unlikely to have been terribly helpful to deaf theatergoers, though, as indeed it is not today - actors roam the stage, emote, and gesture, making lipreading virtually impossible even from the best seats.
So while it is certainly possible for a deaf person to have gone to see Shakespeare's plays, they wouldn't have gotten very much of the content, unless they brought a friend who could convey it to them.
More reading:
The first reply mentions the Emily Cockayne article which is one of the best I’ve seen on the subject. But to add some detail to your question.
It would not have been too difficult to attend a Shakespeare play. Assuming you are seeing the ones premiered at The Globe and not the Blackfriars, as a deaf person (and assuming that you could lip read), you would find the best way to attend was also the cheapest.
As a ‘groundling’ (the customers who paid a penny to enter and did not have seats) you could make your way to the very edge of the stage, allowing you get very close to the action above you. This would make it much easier to lip read. The seats were behind the groundlings and a distance from the stage so standing directly in front was always the best bet.
There was the option that you could sit on the stage if you paid enough but this was mostly only done by the rich wishing to show off. Certainly such a position would leave you vulnerable to cat calls and mockery and as the Cockayne article demonstrates, Elizabethan England tended to treat deafness as a cause of much amusement.
Indeed being deaf could actually be a boon if you could lip read, as the sounds of peanuts crunching underfoot, hecklers, talking, vendors hawking and the myriad other distractions the theatres at the time had would not impact upon you.
Chances are you would not have been distracted by Shakespeare’s rural accent if he was performing either (well known enough for him to mention it in a script for a royal pageant); so overall if you could lip read it would have been a bit of a blessing.
This being said, the crucial element of music would have been missing, but the other senses would be compensated.
Besides afterwards, the wilder nature of Southwark beckoned, and as long as you could see you’d be fine. The only possible issue I see would be if you lived in London not Southwark. Most travelled back and forth via river boat and identification was usually done by sound (the boatmen declaring either ‘Eastward Ho!’ Or ‘Westward Ho!’ depending upon direction of travel they intended to take). As long as you had someone who could hear them bellow, I’m sure you’d be able to arrange your journey home.
Hope that helps.