The Via Valais has two sections with exposure.
The Pas de Chèvre is crossed on stage 2 and requires climbing steep ladders for a short section. While there is air beneath your feet, you are literally on a solid ladder bolted to the wall.
The Schöllijoch is crossed on stage 7 and is the most serious section of the Via Valais. We won't kid you, it is exposed and requires some scrambling between ladders, cables, and handles. It's a rocky, alpine cliff face you need to descend, and while it is maintained as an official Swiss alpine trail, those with a fear of exposure may have problems. Additionally, at the base of the rock face is a fixed rope to access the glacier. Depending on the season, the rope might be long or short.
Once on the glacier, it is more of a snowfield to descend than glacier, with no crevasses, and not steep enough for crampons, as of 2018.
Bonus peaks may involve more technical terrain and scrambling.