I wouldn't put it past the A&E lawyers to insert language in their contract that included a cut of even Duck Commander merchandise sales in stores that are contracted to sell Duck Dynasty goods. But that's as far as such a contract might go.
No fool in their right mind would sign a contract with a television production company that would give that company a share of the profits of said signee's own separate business. By way of analogy, that would be the same as the captains of Deadliest Catch agreeing to give the History Channel a share of their annual crab catch profits.
The chances of them doing that are somewhere between slim, and none.
These people didn't become successful entrepreneurs by making disastrous decisions like that.
They may be hicks, but not rubes.