My sister says she has de Quervain's tenosynovitis. Are family members more likely to get the same thing?

NO. This type of tendonitis is usually associated with repetitive activities that aggravate the tendons. It is not typically considered a hereditary condition. Family members are not more likely to get the the same thing.
Tendonitis. De quervains is a particular example of a tendonitis involving the thumb. It is primarily due to overuse. It is generally not hereditary. The treatment is with Ibuprofen initially, progressing to a cortisone shot if symptoms do not resolve.