A bear's Bite or Claws attacks are considered natural weapons
Looking at the basic rules for monsters' actions, it says this about their attacks:
Melee and Ranged Attacks
The most common actions that a monster will take in combat are melee and ranged attacks. These can be spell attacks or weapon attacks, where the "weapon" might be a manufactured item or a natural weapon, such as a claw or tail spike.
So it seems as though a bear's Bite of Claws attack would be considered natural weapons.
From the basic rules section on combat, melee attacks specifically, it includes the following:
Instead of using a weapon to make a melee weapon attack, you can use an unarmed strike: a punch, kick, head-butt, or similar forceful blow (none of which count as weapons). On a hit, an unarmed strike deals bludgeoning damage equal to 1 + your Strength modifier. You are proficient with your unarmed strikes.
It appears that an unarmed strike cannot be a weapon, and a bear's Bite and Claws attacks are considered "natural weapons", so RAW, the monk's Ki Empowered Strikes would not be applicable to the bear's attacks.
Unofficial rulings (taken from this answer)
There is no official ruling on this issue given in the Sage Advice Compendium as of 2019. Unofficially, rules designer Jeremy Crawford has clearly stated many times on Twitter that natural weapons aren't used for unarmed strikes by default.
For instance, in this tweet:
Ok, Kung-fu panda idea. Moon Druid 2/monk 1. Does a beast's natural 'melee weapon attacks' count as 'unarmed strikes'?
A natural weapon (a claw, horn, bite, etc.) is not an unarmed strike.
A second user interjected, and Crawford responded:
Then explain the following: Alter Form giving you natural weapons that alter unarmed strikes, and Aarakorcas.
Those are exceptional abilities that change the nature of a character's unarmed strikes.
Crawford reiterated this again a few years later:
Am I right that attacks with natural weapons are still considered unarmed strikes rather than attacks with a weapon? E.g. for greenflame blade, silvered fists and similar.
An attack with a natural weapon is not an unarmed strike. An unarmed strike follows the unarmed strike rules in the Player's Handbook, no matter what type of creature is attacking. Some exceptional natural weapons, such as tabaxi claws, can be used for unarmed strikes.
Is a bite an unarmed strike? Do teeth count as natural weapons? Are we starting to go down a rabbit hole? Is the white knight talking backwards?
No (unless you have a special feature). Sometimes (depends on the creature). No. No.
He has consistently stated that attacks with natural weapons are not unarmed strikes, except where the rules clearly state an exception. This is consistent with the rules references above.