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The Clinton camp scrambled to pry loose the white vote Obama had taken in both states.
Literal
Clinton camp [topic-は], both-states [at-で] Obama [subject-が] took white-vote [object-を] pry-apart [volitional-と] became-desperate.
Election-news register. 陣営 (じんえい) literally 'camp, encampment,' is the standard word for a political candidate's campaign staff and supporters — 'the X camp.' 取り崩す ('to pry apart, break down, erode') is a compound verb from 取る ('take') + 崩す ('break, topple'), used figuratively for dismantling something that has been assembled — votes, savings, political coalitions. The pattern [verb volitional] + と + [do/become] is the 'try to do X' construction: 取り崩そうとする or here 取り崩そうと必死になる = 'became desperate to pry apart.' 必死になる ('become desperate, go all-out') captures the effortful stakes.