Sunak said the new approach will be more "pragmatic" and "ease the burden" on families.
A statement from the Speaker's spokesperson said: "If he had the power, the Speaker would recall the House immediately – and he is writing to the Prime Minister to express that view in the strongest of terms.
"This is a major policy shift, and it should have been announced when the house was sitting. Members with very different views on the issue have expressed their disquiet on the way this has been handled, especially as the commons rose early last night, so there was plenty of time for this statemtnt to be made."
He continued: "Instead, the unelected House of Lords will have the opportunity to scrutinise this change in direction this afternoon, when it hears the Government's response to a private notice question on this issue.
"This is not the way to do business. Ministers are answerable to MPs - we do not have a presidential system here.
"The House of Commons is where laws are made, national debates are held, and where statements should be made."
The right-wing of the party is elated at Sunak's plans, saying the Government is "finally listening" - but many MPs have expressed serious doubts.
Senior Tory Zac Goldsmith railed against Sunak, calling for an immediate general election in the wake of growing fury over Sunak's plans.
Goldsmith resigned from the Government earlier this year after the UK's climate advisors warned that they are now less confident in the UK's ability to deliver on its legally binding emissions goals than they were at the same time last year.
Unveiling the details of the new approach this evening, Sunak said: "In a democracy, this is the only realistic path to net zero.
"Consent, not imposition. That's how we'll turn the challenge of net zero into the greatest opportunity and the greatest achievement of our lifetime."
Sunak added: "We are going to change the way our politics works."
The PM promised that the Government would never impose taxes or bans on meat, never force the public to share vehicles, never impose bans on oil or gas drilling and never impose bans on long-haul flights.