No! Mindfulness can be practised just as easily whilst sitting in a chair, standing or even lying down. It is useful to remember that a lot of meditation practices originated in Eastern countries where people are used to sitting on the floor as part of their daily lives. For those of us in Western countries, sitting in chairs is what we are generally more used to and we tend to have less flexibility for floor sitting.
Having said that, during mindfulness courses, I have a range of meditation cushions and meditation stools available so that participants can, if they like, experience sitting a bit closer to the ground!