The vignette is a toll for using Swiss Motorways/Autobahn/Autoroutes/Autostrada. If you cross the border on a non-motorway road, and keep off motorways thereafter, you can avoid buying a vignette.
The side valley from Visp up to Zermatt has one road, which is not a motorway. You just keep going to the end of the road at Tasch, and use the huge ugly car park next to the station, then get a train the short journey up to Zermatt. You will have to pay for parking, and the train. Parking prices are here 15.50 Franks per day.
All railway stations in Switzerland take Euros, including in ticket machines, and I bet the car park does as well. As soon as you spot a cash machine it will be advisable to get some Franks as not everywhere takes Euros, and if they do they give you an inferior rate for the inconvenience to them.