Designs depicting extinct animals is a popular theme among the world's non-circulating legal tender (NCLT) coins - everything from dinosaurs to more modern-day animals, such as the dodo bird, have recently been featured.
"How about among the US commemorative coins?" you ask?
Many different animals were included within the coin designs seen during the classic era of US commemorative coins. Most commonly featured, as might be expected, was the American Bald Eagle - the eagle appears on 11 different coins of the series. Though endangered at one time, the bald eagle did not go extinct and, through diligent conservation efforts, has recovered and been removed from the Endangered Species list.
Other featured animals - horse, moose, owl, dolphin, badger - were/are all doing fine and have no extinction concerns. Several coins of the classic era, however, did feature an animal that is believed to have been extinct at the time of the coin's issue.
Three silver half dollars of the series feature a California Grizzly Bear within their design: the 1925 California Statehood 75th Anniversary, the 1935-36 California-Pacific International Exposition and the 1936 San Francisco-Oakland Bay Bridge Opening issues.
The California grizzly bear was hunted down and killed or trapped due to its frequent run-ins with California's growing human population and the bear's "habit" of killing - for food - rancher's livestock. The California Grizzly Bear was last seen in 1924, the year before the Statehood Anniversary half dollar was released and more than a decade before the other two.
The bear, however, is intertwined in California's history and remains a popular icon/symbol for the state - the animal is even included on the California State Flag! It's not much of a surprise that it would be included on California-themed commemorative coins.
In the "on the road to extinction at the time of release" category is the 1927 Battle of Bennington-Vermont Sesquicentennial half dollar and its catamount. Though the last official sighting of a catamount (eastern cougar) was made in 1938, officials believe that most of its population disappeared during the 1800s due to the deforestation of its habitat and over-hunting. It was thus very close to extinction by 1927. Today, the coin's design presents an officially extrinct animal, as per the US Fish and Wildlife Service.
1925 California Statehood 75th Anniversary
1927 Battle of Bennington-Vermont Sesquicentennial

You can read other of my posts about these coins by visiting:
Commems Collection.