Hello barrenwuffet
Seat reservations on national train services are very uncommon; none of us, the Swiss, would do that and this is why the optional seat reservation is considered a separate step. And, not all trains offer seat reservations. For your itinerary, you can make a seat reservation for the IC8, from Zurich HB to Visp, only. The RE and R trains from Visp to Zermatt do not offer seat reservations at all.
On Swiss trains, there is usually free seating. The trains are seen as mass transit, similar to suburban trains in other countries. As a result, you are welcome to take any vacant seat in your respective travel class. For 2nd class, use the coaches in the front of the of the train (in the train’s travel direction), there are always seats available, compared to the coaches in the center which usually get occupied quickly.
→ If you still want to reserve your seat, kindly note that this is possible for IC, ICE, EC, RJX, TGV, and selected IR services only.
Kind regards,
Roland