I think the answer of jcaron is already pretty good. The official weather forecast service of the government (MeteoSuisse) gives some additional information about the first snow in automn in Switzerland. Unfortunately the page is only available in german, french and italian but not english.
There is no precise information about somewhere in the alps in Valais. But there is some information about Arosa, this is located to the extreme east of the country. Even if this won't be the same in Valais, it is probably quite similar, since the altitude is the most relevant for the snow. For Arosa at 1878 m above the sea, the first snow in automn falls in average the 10th september.
Actually based on my own experience, I think you can expect snow at almost any time of the year above 1500 m. Though, in september and october, it is still quite unlikely under 2000 m and the snow won't stay for a long time.
Your best help will be to look at the weather forecast, they always announce snow falls under 2000 m, since this is relevant for car driving on high routes. If it will have snow at the altitude of a route you plan to drive, just avoid it that day: take another route, wait 1 or 2 day and it will be probably gone at that time of the year.
Also a good information is the website of the TCS (a car driver association), on this webpage, they show the current situation of all passes. You will see if one is closed or if there is some restrictions (e.g. "truck forbidden", "chains mandatory").