For the past thirty years, ad agencies have been dying to get the big tobacco companies as clients.
Literal
This 30 years, advertising-companies [topic-は] major-tobacco-companies [gen-の] customers [obj-を] throat [from-から] hand [subj-が] come-out degree-[ほど] wanted.
のどから手が出る is a vivid idiom literally 'a hand comes out of one's throat' — meaning to want something desperately, so intensely that a hand reaches out from inside you. 大手 'major / big-name' is a common business modifier. ~ほど欲しがった uses ほど to mark the degree: 'wanted to the extent that hands were coming out of their throats.'