Transportation from Zermatt to Murren

www.sbb.ch/en will tell you exactly how to get to Muerren, mostly by train. Last bit from Lauterbrunnen you have a choice, either gondola lift followed by another train, or a bus followed by a 2-stage cable car. The train station and the cable car station are on opposite ends of the town, so choose according to your lodging location.

Surprisingly, there are taxis in Zermatt. They are electric and can hold 5 passengers and luggage. You could also check with your hotel. Many have a courtesy luggage delivery from the station, then you can walk. Not a big town.
https://www.zermatt.ch/en/Media/Listen-Gewerbe/Zermatt-taxi-companies

Edit- I will add there is a slight price difference between the 2 routes to Muerren. on The sbb journey planner, put in Muerren (Schilthornbahn) if you want the bus/cable car route, and Muerren BLM if you want the gondola/train route. I am seeing a price of 50.90 CHF per adult with a Half Fare Card for the gondola/train route, and 53.10 CHF for the bus/cable car route.

In Murren, you walk to your lodging. If the hotels have any type of vehicles, I haven't seen them. Safe travels!

We always walk with our luggage from the train station to our place in Murren. For most people it's doable. If for some reason you can't or really don't want to, Murren does have a couple individuals who transport luggage & maybe a person or two in a really small vehicle, but you have to call to pre-arrange because they definitely don't sit outside the train station or anywhere around town. I don't know that I've ever seen anyone use this service but I keep the numbers on hand. Most everyone just walks. Murren isn't that big.
Stäger Transport +41 33 855 24 80
Feuz Transports +41 33 855 18 70

When you're going to Murren with your luggage, use the route Lauterbrunnen > cable car Grutschalp > train Murren. Way easier than dealing with luggage on the bus in my opinion because when you're down in Lauterbrunnen you can give your luggage to a worker there and they put it on a special part of the cable car that then transfer for you to Murren. So once you get to Lauterbrunnen you hand off your luggage for the rest of the journey. You pick it back up in Murren on the open air "trailer" attached to the back of the train. They don't tell you about this, it's just sort of assumed that you know. When you're entering the cable car station in Lauterbrunnen (across the street from the train) you'll see 2 escalators that take you up to the cable car. BEFORE you go up the escalator, veer to your right and you'll see a loading area. Flag down a worker and show them you have luggage. They will either show you where to put it or load it up themselves. It rides in a carriage under the cable car with you and then they transfer it over at the top onto the train trailer. You definitely want to do this because the train to Murren is only 1 car and it's almost always full so there really isn't room for everyone to carry all their luggage inside the train. You can keep your personal carry on with you.

In Zermatt, there are a ton of e-taxis. It's actually a bit annoying, because Zermatt has gotten so packed with tourists that those taxis are just flying around everywhere and kind of negating the whole "car free village" vibe. Zermatt is much much larger than Murren so depending on where your lodging is I'm sure it's advantageous to use one. They just sit lined up right outside the train station. TONS of them.