Most if not all taxis in Bienne will accept credit cards (VISA, MasterCard, Maestro and PostCard). There is no Uber in Bienne, Switzerland is not very well serviced by Uber.
As far as apps go, I only tried "Metro Taxi Group" which had not a lot of drivers on it. Usually you order a taxi by calling a central number (which you can find almost everywhere). If you call for a taxi, specify that you need to pay by card, same if you hail one, ask if they take the credit cards before getting in to avoid bad surprises.
Bienne is not a big city and the bus service is pretty good, so you'll find most destinations are in walking distance or within a short bus ride.
Have a good stay in Switzerland !