Cabinet Doors

Will you be having these new doors fabricated or will you be making them yourself? If you don't have the appropriate tools and comfort and skill levels to get the job done hiring someone else might be less expensive than trying it yourself, buying a bunch of tools, and then leaving them to rust.

There are a few Carvers in my part of the world though I can't say that I ever recall being inside of one long enough to notice the cabinets. What you describe sounds like a tambour door, which is a lot like a roll top desk, with a dado/channel/track in the top and bottom and a bunch of slats that make up the door. Presumably it curves back into the body of the piece to get the door out of your way when it's not in use. Usually the first slat is heavier than the others to support the handle. Was it this slat that broke, or multiple slats?

Some detail photos would help.

If you are not too worried about color matching the easiest way to effect a repair on what I think this cabinet probably looks like is to cut out the damaged slats. Then mill a piece of flexible (relatively) wood like poplar or white oak to size, give it a quick steam, and bend it and pop it into the track. Then use a contact adhesive to adhere a new layer of fabric across the repair and several slats to either side of it. Be careful not to get any adhesive of any sort near the track, as that leads to resistance and frustration, and a combination of those two factors is probably what necessitated the repair to begin with.

If wedging one in there isn't an option (be sure to investigate coming in from the back with the slat, or laying the slat down on its side in the track and then standing it up) you can scarf two pieces together, joining them with a long tapered joint, which will be fairly visible if you aren't exceptionally careful in matching the grain.

If neither of those options appeal to you the remaining option may be to start taking the whole piece apart. Keep in mind that if you don't know what you are doing nothing that is made out of wood will look as pretty after it's been taken apart and put back together. If you luck out the dado/channel/track is made out of a rabbit in the body of the piece with a bit of trim affixed to the front, and a broad knife, a small hammer, and a bit of elbow grease can get that taken off and get you the access needed to remove and repair and replace the door. Or it could be affixed some other way.

As before, detail pictures would help immensely to figure out just exactly what kind of problem you are facing.