No, we cannot do a formal Ardaas (a humble request) sitting down. The Maryada (system of discipline) is to do Ardaas standing up because we are present during that period. Some people may be lying down or sleeping. But the ones who stand are present in Ardaas. For example, at the time of marriage, the people partaking in the Ardaas stand up. The couple and their parents stand up in Ardaas. They are accounted as present in the Ardaas and if the whole Sangat stood up during that Ardaas, they would all be accounted as present too. So it is important as it is the Maryada. 

You can watch Bhai Jagraj Singh explain this in the video below: