No, it isn't necessary for an article to appear each time. However, the two sentence structures have different connotations.
I saw a bear, a lion and a donkey.
connotes that, at various times, you saw a bear, a lion, and a donkey.
I saw a bear, lion and donkey.
connotes that you saw a bear, lion, and donkey together in a group.
This connotation isn't inescapable, but it would be the default understanding.
There is no article before yearling or cub in your example because the writer intends to describe the three as appearing together. The construction also draws on an established understanding of a mother bear, yearling and cub being an expected family unit for bears to be traveling in; the entire construction is, in a partial sense, being treated as a single collective entity.