He lent it his name. It was illegal for people to be itinerant actors, that is, actors who could not prove that they worked for someone. The Lord Chamberlain very graciously allowed the actors to say that they were the servants of the Lord Chamberlain, so they would not be arrested as vagrants. That was the extent of his involvement.
They were a company of actors.
The Lord Chamberlain who was the patron for Shakespeare's acting company was Henry Carey, the 1st Baron Hunsdon. He served as Lord Chamberlain from 1585 until his death in 1596 and supported the Lord Chamberlain's Men, the acting company to which Shakespeare belonged. Under his patronage, the company flourished and performed many of Shakespeare's plays.
They didn't change their name--it was changed for them when their patron changed or when his job changed. In fact their name changed several times: From the Lord Chamberlain's Men to Lord Hunsdon's Men when the Lord Chamberlain died and his son Lord Hunsdon took over patronage of the company, from Lord Hunsdon's Men to the Lord Chamberlain's Men when Hunsdon became Lord Chamberlain like his father, and from the Lord Chamberlain's Men to The King's Men when King James I took over patronage of the company. The partners in the company had nothing to say about it.
Shakespeare did not join a different company. The Lord Chamberlain's Men merely changed their name to the King's Men. The company remained intact.
The Lord Chamberlain's Men
He became their patron.
It was a partnership or company of actors formed in 1594 under the patronage of Lord Hunsdon, the Lord Chamberlain and which included among its members Richard Burbage, Cuthbert Burbage, William Shakespeare, Will Kempe, and Augustine Phillips.
The name of every Elizabethan and Jacobean acting company tells you the name of the company's patron or sponsor. It's like having an acting company called the Coca-Cola Players. Lord Strange's Men, The Lord Admiral's Men, The Lord Chamberlain's Men and the King's Men were sponsored by Lord Strange, the Lord Admiral, the Lord Chamberlain, and the king, James I, respectively.
Lord Hunsdon was the Lord Chamberlain of England; so the company took his title as its name.
The theatre company to which Shakespeare belonged for most of his career started off as The Lord Chamberlain's Men, but when their patron changed from the Lord Chamberlain to the King their name changed to the King's Men.
He was an actor, writer and co-owner at Lord Chamberlain's Company...
The Lord Chamberlain's men.