yes you can. I personally would prefer to replace the door with two drawers, the top one would hold the rubbish bins, the bottom one something else. I don't think to have the bins at the bottom of the cabinet is the best solution.
But, what you could do without any big changes is a pullout bin which is fixed inside the cabinet and slides out. However, this means, you have to open first the door fully and then pullout the bin drawer. Whereas with only drawers it is easier and quicker. At the bottom of this cabinet you could install a pullout drawer for something else.