Jeremy E Crawford says here that (emphasis added by me):
The intent is that the druid uses the bonus in the beast's stat block for ANY proficiency the druid LACKS.
Meaning if the druid has the proficiency he can use it.
Weapon Master says I can choose four weapons to gain proficiency with.
If I choose claw, bite, sting and slam, would this allow me to add my proficiency bonus instead of the animal's to my attack rolls?
Are they weapons I can pick for the Weapon Master feat?
Best Answer
No.
Weapon Master (as amended in the PHB Errata):
The natural attacks of beasts are not made using simple or martial weapons.
In any case, the game statistics of a druid using the Wild Shape class feature are explicitly stated to be that of the the beast plus a limited list of the game statistics of the druid's normal form. That list, as other answers spell out, doesn't include weapon proficiencies.