I think your best bet is to get milk from cows in alpine pastures. There are certainly cows up there that are supposed to eat grass. Not sure if it will be grass only, but I would expect so, as summer grass should be sufficient feed (but am no expert!). Basically, they produce cheese from these cows, which is called Alpkaese and which involves a protected manufacturing procedure (has to come from above 1500m and from pasture fed cows; bergkaese is not the same as it can be produced in high areas at any time, and not just from grass fed cows, see below).

So I would find out from perhaps the organisation of farmers that produce Alpkaese if the milk from these cows is also sold separately. Maybe start here? http://www.schweizeralpkaese.ch/

Bergkäse und Alpkäse wird oft unter demselben Namen verkauft. Alpkäse darf Bergkäse genannt werden - umgekehrt gilt das nicht. Während der Bergkäse auch während des Jahres am Hof erzeugt wird, wird der Alpkäse ausschließlich im Sommer erzeugt, wobei die Kühe fast den ganzen Tag im Freien verbringen und sich von den Alpwiesen ernähren.
See also http://www.chaeschilbi.ch/bergkaese.html