She voted for Mr. Nishioka regardless of party affiliation.
Literal
She [topic-は] political-party [regardless of-に関係なく] Nishioka [object-を] voted-for.
政党に関係なく means 'regardless of political party' — に関係なく is a set phrase meaning 'without regard to, regardless of.' 投票する (とうひょうする) means 'to vote.' 西岡氏 (にしおかし) uses 氏, a formal title equivalent to Mr./Ms. used in news and formal writing.