It is quite common for companies to require that their employees have at least one "reliable" form of transportation (personal vehicle, public transit, bicycle, and/or by foot, etc).

Companies also have a right to expect that you clock-in on time while being in a clean and properly dressed condition. That said, I very seriously doubt that any company would, or could, tell you that you can't ride your bicycle to work as long as you meet the above conditions (like every other employee).

As for the "showing up while drenched in sweat" problem; your company might give thought to installing showers and dressing rooms. Otherwise, you'd need to make other arrangements (as mentioned in previous posts).