As the preceding pictures show, no-flash photography in a gym shouldn't be a problem with fast lenses such as a 2.8 and faster.
My concern is white balance. Lighting varies from sunlights, incandescent, florescent, sulfur-vapor (orangish), mercury (bluish), and these days, halogen (still quite expensive at present). Each of them can easily 'trick' the auto-white-balance in any camera, in my estimation.
Definitely get familiar with using a white balance card and setting Custom White Balance in your camera for each gymnasium, and perhaps each time of day, if there is a large number of windows/sunlights that could affect color balance. If you shoot in RAW format, you can easily do white balance correction in post processing rather than in-camera using CWB. But you'll still need to take a picture of a white balance card in RAW format to set the balance in post processing.