If I wildshape into a bear form and have the Charger feat (I'm a human druid with feat variant), can I then use Charger, charge as a bear, use the attack action granted to take a bear's multiattack, then get +5 to damage for each attack from the bear?
No, you can't use multiattack on a charge.
The Charger feat, from page 165 of the PHB:
When you use your action to Dash, you can use a bonus action to make one melee weapon attack or to shove a creature.
If you move at least 10 feet in a straight line immediately before taking this bonus action, you either gain a +5 bonus to the attack’s damage roll (if you chose to make a melee attack and hit) or push the target up to 10 feet away from you (if you chose to shove and you succeed).
Multiattack is an action that allows you to make multiple attacks. This does not mean you can substitute it into anything that gives you an attack. The Charger feat lets you make one melee weapon attack as a bonus action, and that's all you can do with it.
In the case of a Druid in bear form, you could make one Bite attack OR one Claws attack OR one shove as a bonus action after you used your action to Dash.
To be able to use multiattack, the Charger feat would have to say something like: "When you use your action to Dash, you gain another action."
For more information on bonus actions, check out page 189 of the PHB, or page 69 of the Player's Basic Rules. For more information on multiattack, check out page 11 of the Monster Manual, or page 6 of the DM Basic Rules. Alternatively, there are some questions on this site about both bonus actions and multiattack.