Police Federation chairman Jan Berry said there was a sense of "betrayal" when agreement was not reached over pay.
She told BBC News: "Because we don't have the right to take industrial action, we have arbitration.
"Arbitration is binding on the police officers and it wasn't binding and it isn't binding on the home secretary.
"Whilst we've had 28 years of being able to trust the government to honour either the arbitration finding or the agreement of the Police Negotiating Board, last December they decided to betray that trust."
Ms Berry added it was not a vote for strike action, but a go-ahead to lobby the government on the issue.
The vote seems to be a deliberate shot across the bows of a government which many Police Federation members have come to regard as hostile
In January, officers decided to seek the views of their colleagues following a mass rally in London.
More than 20,000 officers marched on Westminster following Home Secretary Jacqui Smith's pay decision.
Police officers were asked whether they should lobby for the right to strike - in addition to other industrial rights - if ministers failed to abide with pay arbitration deals.
Work and Pensions Secretary John Hutton, whose remit includes industrial relations, said he believed it was unlikely that police officers would strike.
"I don't think there's a police officer in the country who wants to strike and let's hope we never get anywhere near to that position," he said.
Keith Vaz, Labour chairman of the Commons Home Affairs Select Committee, said he fully understood why the police had voted for the action.
He said: "I urge the home secretary to immediately enter into a dialogue with the leaders of the federation to ensure that the necessity for them to take such action is considerably diminished, if not eliminated."
Shadow home secretary David Davis said the government had been "dishonourable" in its behaviour towards the police.
The Conservatives have proposed an arbitration system which could only be overruled by the House of Commons, which Mr Davis said would be "fair, transparent and serve the public interest".
The police are currently banned from going on strike or taking other industrial action, and any such action is a criminal offence.
At their conference, members of the Scottish Police Federation rejected the right to demand full strike action but voted narrowly in favour of seeking other industrial rights.
There was less anger about pay levels in Scotland as police officers had their 2007-8 pay rise backdated to September 2007 giving them the full benefit of the 2.5% increase.
Colleagues in other parts of the UK had their increase backdated only to December.
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