As @jphi618 indicates in his comment, glass or sheet acrylic would be thin, fairly strong and fairly cheap, then painted to match the door. There is also sheet steel, which would be much heavier and expensive, but similar in thickness.
There are other materials, such as sheet MDF and plywood that are available in 1/4 inch thickness and hardboard that is available in 1/8 inch thickness. These are all thicker than standard window pane glass, but the glass in your door may well be double thickness, which is around 1/8 inch.
Plywood and MDF are fairly easily cut with power tools, as is hardboard, but the latter is tougher. All of them can be painted.
If you use something other than glass of the same thickness, you probably want to adjust the moldings to accommodate the additional thickness. You also may want to replace the other panel so they look similar. This also eliminates the chance of the other panel breaking.
The danger of painted glass, especially if it is not thicker than window pane, is the risk that people push on it believing it is wood or some other non-fragile material. People tend not to push on unpainted glass unless it is obviously very thick.