The availability of the Oz books in electronic form makes this sort of search quite easy. I did a quick search of all the Oz books I have and found lots of references, though admittedly many of the hits were for the word "bear" as a verb.
Anyhow rather than spend ages going through all the search results I stopped at the first mention of a bear. In book 4, Dorothy and the Wizard in Oz, there are invisible bears that become visible when killed:
As the little Wizard turned to follow them he felt a hot breath against his cheek and heard a low, fierce growl. At once he began stabbing at the air with his sword, and he knew that he had struck some substance because when he drew back the blade it was dripping with blood. The third time that he thrust out the weapon there was a loud roar and a fall, and suddenly at his feet appeared the form of a great red bear, which was nearly as big as the horse and much stronger and fiercer. The beast was quite dead from the sword thrusts, and after a glance at its terrible claws and sharp teeth the little man turned in a panic and rushed out upon the water, for other menacing growls told him more bears were near.
I didn't find any mention of Tigers apart from the Hungry Tiger.
The events of book 4 take place in the Land of the Manaboos not in Oz. There is an indirect mention in book 5 The Road to Oz:
"I am not so sure of that, sir," answered the Tin Woodman. "A while ago the crooked Sorcerer who invented the Magic Powder fell down a precipice and was killed. All his possessions went to a relative – an old woman named Dyna, who lives in the Emerald City. She went to the mountains where the Sorcerer had lived and brought away everything she thought of value. Among them was a small bottle of the Powder of Life; but of course Dyna didn't know it was a Magic Powder, at all. It happened she had once had a big blue bear for a pet; but the bear choked to death on a fishbone one day, and she loved it so dearly that Dyna made a rug of its skin, leaving the head and four paws on the hide. She kept the rug on the floor of her front parlor."
Alternatively in book 11 The Lost Princess of Oz there are several bears, though I'm not sure if this is strictly speaking in Oz.
Aha! Book 13 The Magic of Oz takes place in Gugu Forest, which is said to be:
the home of most of the wild beasts that inhabit Oz
and therefore presumably in Oz. In this story we meet Bru the Bear.