The Vinsmoke Family once dominated the North Sea by force; even though they've declined now, they remain a force to be reckoned with.
As for why Doflamingo and Big Mom's deal would involve the Vinsmoke Family, it is entirely Doflamingo's deliberate scheme against the Vinsmoke Family.
Doflamingo is willing to hand over the method to manufacture artificial fruits, but that means nothing to Big Mom. Even if she possesses the method, she wouldn't be able to produce artificial Devil Fruits, unless she kidnaps Caesar Clown.
If Big Mom does kidnap Caesar Clown, it would undoubtedly reignite war, and at that time, the main force of the war wouldn't be the Donquixote Family, but Kaido.
Therefore, acquiring the method to produce artificial Devil Fruits is useless to Big Mom, yet Doflamingo presents her with a promising opportunity.
