Either take a couple pics and post them, or remove one handle + cartridge or stem and take them with you to a plumbing supply to help identify. (Make sure to take the handle and button, if any.)

I won't be pulling a cartridge until I give up finding any info on the manufacturer because of possible damage (both stems are broken inside). The logo is too small to photograph, thus the description. Both faucet knobs are missing the end caps that cover the screws that fasten the knobs to the stems, so that possible ID is unavailable. I guess I am stuck with taking pictures and hoping that I can find a knowledgeable plumbing supply.

Just take a couple photos of the whole thing and post. We may recognize just from the handles. Even that tiny logo can probably be snapped with the macro setting on a digital camera...or use your magnifier.

Here are some pics. The trademark is a tiny dot at the top of the inner ring on the faceplate of each faucet handle. The following picture in the next post is the actual trademark. I found a supplier who thinks they were made by either Valley or Eljer, and he sold me parts to fix it if he is correct. I'll find out in a day or two if he was right.

This is the actual trademark, I had to hold a magnifying glass up to the camera to get this one. It is circular, not oval, apparently the picture is being distorted by whatever software is being used by this board.

Not sure the brand but it doesn't make a difference here. That doesn't have cartridges - it has STEMS that can be removed for repair. You need to shut the water off, remove the handle, remove the collar, trim, and stem. Then take both of them to an old fashioned hardware store. There, you will purchase and replace both the seat washer and the screws. (Do it there so they can help you match the sizes.)

Here's the catch. The seat screw (which holds the washer) may crumble when you try to remove it. Don't worry. If so, you will then have to buy a tap to remove it. Ask your old fashioned hardware store to walk you through it. (have your cordless drill and drill bits with you).

You should be able to leave with stems that are ready to go. Reinstall snugly and you'll be fine.

Good luck.

BTW, you'll also need a plumber's socket set to grab the stem and wrench it loose. Here's a link to see them.

Here is a link that might be useful: plumber's sockets

One last thing. This is a one visit repair by a good plumber, whereas you'll put a morning into this. Might want to just get a plumber and watch how it's done the first time.

I wish it was the old rubber washer and stem setup, that is the only type of faucet I ever repaired... until today. The picture is what I removed after unscrewing the collar. The plumbing supply guy was correct, it was a Valley brand cartridge, so the replacements he sold me were correct. When I removed the cartridge, I had to pry out the black rubber ring and spring. The white nylon insert for the spring was inside the rubber piece on one faucet, but it was in the other end of the spring on the second faucet. I called the supplier and he was uncertain which way was correct, but we reasoned that there was more need of the insert on the rubber side to help prevent the spring from cutting the inside of the rubber ring. His knobs and adapters were different, but the stops on the old adapters were broken anyway (they turned through 360 degrees) so I did a complete facelift on the fixture, only leaving the trim plates, so now I need to buff off that corrosion on the tub spout. The replacement parts did not include the handle screw for some reason. I went to the closest traditional hardware store (no big boxes there) to get the original screws (each screw was different when I took the old handles off so I suspected that neither was original). The hardware store had only one of my cartridges, and an employee finally found a Valley parts listing, tried to look up the proper screws but they were listed as not available. The inside of the new plastic replacement stem is unthreaded, so apparently you are expected to use an appropriate size and length sheet metal screw, so I bought a couple of SS screws to mount the new handles. I paid a premium for parts from the plumbing supply store, but the advice was well worth it, and saved me from calling a plumber for what turned out to be a fairly simple job after all.
The hardware store I visited to get replacement screws did not have employees who were very knowledgeable about plumbing. They admitted that their fairly large selection of repair parts for faucets was an orphaned area in the store. I suppose that with the variety of repair parts and manufacturers, some of which seem to be out of business now, it is fairly easy to just throw in the towel and buy new fixtures now.
I will probably be doing this same fix soon at my house. My fairly expensive delta faucets in my bathroom have started leaking already after less than 5 years in service, and I know they will take cartridges too. Quality is being sacrificed for cost... same with lighting IMO, as I see import stuff selling for hundreds of percent above what these fixtures are actually worth when you look at their construction and quality.

Here is a link that might be useful: Valley faucet cartridge replacement