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.
You can get a trials bike. They often do not have a saddle or seatpost, by design.
They are meant for doing tricks, jumps, balancing, etc. This doesn't sound like what you are doing but if you must not have a saddle, or post, this might be an option for you. This is what they look like:
You might want to just use a regular bike and stand up instead as it will probably ride better for normal riding while standing.
Some options which existed before, but apparently disappeared. Both are essentially a fold-down scooter with pedals+chain.
- Occam Cycle http://www.greatscottech.com/
- Micro pedalflow http://www.micro-mobility.com/en/products/micro-pedalflow-white
Pedalflow is still available in some countries for about $400 USD.
Impressions: The small front wheel and high center of mass imply a particularly front-tippy ride, should the rider find a pothole with the wheels, or otherwise have to make a quick stop. The Occam cycle raised about a third of its kickstarter target, and was not funded.