There is a reason that it is difficult to get a reservation at Chez Vrony. The view and the food are fantastic. Absolutely make a reservation here.
TIPS for NON-SKIERS. It took me several asks along the way to find Chez Vrony. Its a about a 30 minute HIKE down to the restaurant. You have to have hiking boots (and maybe a hiking stick) if its really snowy you may want those grippy bottom things for your shoes.
Give yourself an HOUR to get there from Zermatt Village
You have to take the train from the Zermatt Sunnegga Station (you need a ticket for this / or your ski pass) you take this Awesome underground train up the mountain to Sunnegga and exit there. SKIERS - will simply ski down to Findeln (honestly, this is the best way to get there, but if you don't ski keep reading) NON-SKIERS (about 30 mins hike from here) - when you exit the train you will see a tunnel to the kids adventure area - walk through the tunnel. Take the small lift down to the kids area, it actually a very nice view, then exit on to the snow. You are going to walk about 100ft along the side of the ski slope and on your left you will see a HOT PINK POLES with a trail entrance - Follow the poles down the trail. You will pass through this little Hotel/village looking place - there will be PINK poles and arrows directing you to Vrony. Just follow the pink poles and trail.
You will cross a ski run/slope - walk straight across - follow the poles. You are about 5 mins away at this point. Good Job!
You will see Vrony flags and the restaurant.
OK - Now to get back. I went back the same way I came down, mostly because I knew the path and so I felt sure of myself. It is uphill which was a hall, but since I was not skiing that day (huge mistake on my part) I wanted the exercise. You can ask the staff at Vrony and they will advise on alternative options like hiking down to a chair lift and taking that up to Sunnegga. You have to take the underground train back down to Zermatt.
OK. Real quick on the food. YUM!!! Get the cheese fondue, its so fun, and warm, and Swiss (they have gluten free bread if you ask). My kids had the One and Only Vrony Burger - it was delicious and it should be for 40 francs.
The view is amazing and the Matterhorn, is well, majestic.
Hope this helps. I should have taken more photos of the path, but I didn't think about writing this till it was all over.
Enjoy Zermatt.