Does anyone manufacture a seatless bicycle? I need to stay fit and cycling is a great non-impact exercise. I don't see the point in sitting when cycling, instead I rather standing as such as it creates a better form of exercise. Are there any bicycles that are designed without a seat specifically for this purpose?
Most trials bicycles are designed without a seat.  However, they are likely to be uncomfortable when ridden for long distances.

You may want to consider what what you mean by "stay fit".  Fitness must have goals to be obtainable.  Cycling is generally consider to be cardio exercise that improves aerobic fitness.  Standing while cycling is more of a strength training exercise that is less maintainable over time.  I don't know of any manufacturers that produce a bicycle meant to be stood on all the time and intended for distance.  It's not something people try to do often, so if their is one, it would be very small, selling to a niche market.