How can I stain a door to exactly match this kitchen cabinet door?

Hello Jeanne! You don't mention whether the door you are staining to match is raw wood or whether it already has a finish on it. You can take one of your cabinet doors off and take it to your local home improvement store to try and match the stain color. You should be able to get fairly close if not exact. You should also finish your door with a few coats of poly. You can buy poly in the sheen that matches the cabinets if that is something you'd like to consider.

If you have no experience in woodworking take a door to the homecenter kitchen cabinet department. They may be able to match something close. As a woodworker I would have no problem duplicating the door with the tools I have. Another option would be take a door to the flooring department and look for laminate flooring that looks close. You can glue the tongue and grooves, glue/nail two or three thin battens on the back to hold the panels together. Use black/brown paint or black/brown Sharpie for the grooves.

I would take a door to a hardware store for wood type and stain advice. It looks like paneling in the photo. My sister has something very similar and she ended up staining all the cabinets darker as well as removing a few doors when she had an issue.