I'd suggest booking an apartment instead as many will offer use of laundry facilities. Hotels mainly offer an overpriced "laundry service" rather than self-service machines, whereas most apartments will have access to a laundry room.
Hotel Bahnhof does have a coin-op laundry, as does the youth hostel, but both of these are on the budget end of the scale.
In town, there are a couple of places (google Waschsalon Zermatt - Waschsalon Doli gets good ratings) that will do laundry for you, but I'm not aware of any self-service laundromats. These are fairly rare in Switzerland in general. Where will you be before/after Zermatt? There are self-service laundry places in Luzern, Interlaken and Bern for example.