What a great idea!
Here are a few from Washington state:
Hurricane Ridge (Olympic National Park):
North Cascades (Newhalem Visitor Center):
Mount Rainier (Paradise):
Here are few others:
Glacier National Park Webcam:
List of National Park Webcams:
The national park site does not contain an exhaustive list of national park webcams (it doesn't list the Glacier webcams, nor does it list the Hurricane Ridge webcam but it does have a different webcam for the Olympics). Go figure.