People who lose their hearing as adults, at pretty much any point in history, continue their current job with modifications. This is even more true in the Middle Ages, because the jobs don't rely on electronic technology that deaf people are sometimes not able to use (phones, radios, etc). So they'll continue to be a butcher, weaver, etc. They'll probably continue speaking for themselves, and develop their own set of gestures to communicate with customers, colleagues, etc. There's some interesting notes about adult-onset deafness in Ottoman Syria in the 16th-18th centuries here, and of course we have the well-known examples of Edison, Beethoven, etc.

People who were born deaf in the Middle Ages would not have access to education; the earliest record of deaf education places it in the 16th century. But we do know that it was common for deaf people in this period to use gestures with those around them, as in medieval Iceland. We don't know of signed languages being developed at this period, as those require deaf people to be in one place to develop a full language (which did not happen until education started). But people used gestures just fine. They would not have particularly skilled jobs due to the lack of education, but they could do pretty usual tasks, such as farming, baking, etc.

The Justinian Code's view of deafness was not kind to pre-lingually deaf people - if you couldn't speak, you were pretty much treated like a child. Metzler's great Disability in Medieval Europe even talks about deaf people being let off for crimes because they couldn't understand what they had done, as we sometimes do for young children today.

But deaf people were nonetheless integrated into medieval society. A great reference for this is actually Chaucer's The Canterbury Tales. The Wife of Bath, one of the most popular characters, is deaf. She became deaf later in life, but she has a job, romantic relations, etc. See more here on the Wife of Bath's deafness.

For more reading besides the links above, see Aude de Saint-Loup, "Images of the Deaf in Medieval Western Europe," in Looking Back: A Reader on the History of Deaf Communities and Their Sign Languages, Renate Fischer, ed.