Natural attacks are a sort-of gray area between general unarmed attacks and weapon attacks. However, a careful reading of the Tavern Brawler feat reveals that it says (emphasis mine):
- When you hit a creature with an unarmed strike or an improvised
weapon on your turn, you can use a bonus action to attempt to
grapple the target.
So, clearly having a weapon in hand is not a limiting factor. Moreover, when the MM was describing natural weapons it said:
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.
In which case, I would assert that the book is trying to define two categories of attacks: spells versus weapon. And in that case, natural weapons falls under the 'weapon' category. This is important because spells will use Intelligence, Wisdom, or Charisma for their attack rolls, whereas weapons use Strength or Dexterity. It is not meant to be an assertion that natural weapons are armed attacks (like with a sword) against unarmed attacks (with a fist).
I would check with your DM, of course, because he or she may have a different take, but I'd say the rules as written would definitely allow a shape-shifted druid to take advantage of both natural weapons and Tavern Brawler feat.