Mountain goats are not native to Yellowstone National Park. Mountain goats (Oreamnos americanus) were introduced in Montana during the 1940s and 1950s. They established a population in the park in the 1990s and have reached a relatively high abundance in the northeastern and northwestern portions via the Absaroka and Gallatin mountain ranges. There are about 208 mountain goats in and adjacent to Yellowstone.
A mature male (billy) weighs 300 or more pounds; female (nanny) weighs 150 pounds. Their young (kids) are born in late May–June. They have white fur. Both sexes have horns; females horns curve less and are thinner and sometimes longer than males. They live in very high, rocky and precipitous terrain.
There is one spot I go to view mountain goats from the road, which has been extremely productive every spring. On the Northeast Entrance Road, heading toward the Northeast Entrance, there are two pull outs on the right hand side of the road shortly before the Pebble Creek Campground turnoff. Looking across Soda Butte Creek and up into the rocky tops of closest mountain, look for bright white spots. Unfortunately, this spot requires binoculars and/or a spotting scope and a high power zoom to get a good look at the goats. Hikes up into the mountains have a better chance of encountering goats.
There are better (easier) places for viewing Mountain Goats: Glacier National Park (Montana); Jewel Basin area of Flathead National Forest (Montana); Mountain areas around Bozeman and Missoula, Montana; and Olympic National Park (Washington) are areas I have had much closer viewing opportunities.