Sorry, pun intended. I haven't seen your actual cabinets, so keep that in mind.
Many cabinet doors are constructed of a solid wood frame, which is usually cut in a decorative profile, around a 1/8" furniture grade plywood sheet, which is stapled or screwed to the back of the frame. Ideally, you should be able to pop that plywood out without destroying the frame in the process.
Before you do, run a blade around the inside of the frame on the front, to cut the clear finish that probably built up in the crack where the ply meets the frame. This will help loosen the ply and avoid some damage to the frame.
Once you get the ply out, you should be able to locate a piece of TEMPERED glass that is the style you want...(clear, frosted, wire reinforced, dimpled, etc) and have it cut to fit the door. Make sure you specify the glass to be smooth finished on the edges so you don't slice yourself up.
You should be able to attach the glass to the back of the doorframe with some screw on clamps. If you describe the project to a "hardware" guy, he should be able to point you in the right direction.
Make sure you predrill your holes in the frame before you insert the screws, you don't want split wood at this point, and don't penetrate the face of the door.
That's it in a nutshell. If you find you can't do it this way, you may be able to actually order doors like this from the company who made your cabinets, if they are still in business. Most companies will leave a sticker on the inside of a door or drawer.