One proposed design for a Dyson sphere is the Dyson bubble, which consists of statites (satellites that use light pressure to counteract gravity). In this design, the statites remain stationary relative to the star, but they need to be very lightweight, with an average areal density of 0.78 g/m2 (about 1/100 the density of paper).
However, would a "hybrid" design work, one that uses a combination of light pressure and orbital velocity to counteract gravity? This seems to be possible according to the Wikipedia page for statites. With this idea, the statites wouldn't have to be as lightweight as "pure" statites. Also, if my understanding is correct, this would still allow people living on the statites to experience gravity, something not possible with a regular satellite (unless it's rotating, which introduces its own issues). Of course, this means the statites would now have to move relative to the star, which removes some of the Dyson bubble's advantages.