The government says the rights the charter protects are already covered by British law and May's spokesman said earlier on Monday the withdrawal bill, which was approved by the House of Commons before being sent to the upper house, was the best way of providing "the smoothest possible Brexit".
It is a trading bloc made up of the EU's member states, in which goods move freely across borders without tariffs.
The current system allows EU countries to apply the same taxes on imports to goods from outside the union.
It reduces administrative and financial trade barriers, such as customs checks and charges, and boosts economic co-operation.
Any trade deals with countries outside the bloc are agreed centrally with Brussels.
Although the vote was not binding, it secured a new debate in the House of Commons, where a number of MPs are expected to back the position.
The vote is on an amendment to the EU Withdrawal Bill, which the Lords backed last week, to keep the option of a customs union on the table.
Johnson has previously said that staying in a customs union would be "worse" than remaining in the EU. Leave voters were more likely to say they would not vote in the same way now, with one in 13 Leave voters saying they'd act differently (8%) compared to one in 29 Remain voters (3%).
Among Leave voters, just 9% think Britain is better off economically in Europe, while almost a quarter of these voters (23%) think we should continue to be part of the Single Market.
"It doesn't mean we're coming out of Europe but it does mean that we need to change our relationships with the institutions of the European Union", he said.
But pro-Brexit politicians have denounced the move as an attempt to derail legislation created to reflect the popular will expressed in the June 2016 referendum.
Senior euroskeptic ministers are likely to make their case for breaking free from European Union rules directly to the prime minister on Wednesday, when she holds a key meeting of her inner Cabinet focused on the Brexit negotiations, Tim Ross reports.
The British Government remains committed to leaving the EU Customs Union according to sources in Downing Street. One of May's senior aides told a meeting last month that her team would "not be crying into our beer" if Parliament forces the Government's hand.
Shadow foreign secretary Emily Thornberry also told the Marr show staying in a customs union "makes sense".
The opposition Labour Party wants to stay in a customs union.
The Irish border is a key element of their case.