MPs and political campaigners claim voters have been turned away from polling stations for not having the correct identification.
One voter said she cried as she went to vote after seeing an elderly woman turned away for having the incorrect ID.
For the first time on Thursday voters in England must show photo identification in order to cast their ballots.
Critics of the move, instigated by the Conservatives, say it will deter young people and ethnic minorities from voting.
Lib Dem MP for Oxford West and Abingdon Layla Moran said she was worried “significant numbers” would be unable cast ballots because of the new rules.
She tweeted: “We’ve had reports by our tellers of people being turned away at polling stations for lack of correct ID.
“That’s just in my constituency so far. Across the country I’m worried this will be significant numbers and far more than the exactly 0 people found guilty of fraud last year.”
Tory MP for South Thanet Craig Mackinlay hit back telling her: “Really? Polling stations in South Thanet I’ve visited have had zero turn-aways. Some gripes about it for sure but poll card delivered to each elector has clear details.”
One social user, called Tor, tweeted: “Cried at the polling station this morning as the old lady in front of me, who had struggled to walk there, was turned away.
“She had photo ID but not the right version. When I handed them my passport, they questioned if the photo was me. Horrible atmosphere.”
Labour MP Lloyd Russell-Moyle added: “Early reports back of voters being turned away from voting for not having ID. Please remind your friends and family voting today that they need ID to vote!”
Megan Kenyon, journalist for the Local Government Chronicle, said she heard reports of people trying to use their firearms certificates as ID.
She wrote: “Very random - I’ve heard reports from two separate councils of residents showing up to vote hoping to use their firearms certificate as voter ID and being turned away because FACs aren’t on the list of valid ID - they don’t have a photo!”
Conservative MP for Southend West Anna Firth admitted she had to go back to the car to retrieve her ID so she could vote.
The Electoral Reform Society, which has strongly opposed the introduction, urged ministers to rethink the new law.
But the Association of Electoral Administrators said the polls were “running as smoothly as usual”.
Jess Garland, the Electoral Reform Society’s director of policy and research, said: “We’re already seeing countless examples of people being denied their right to vote due to these new laws.
“From people caught out by having the wrong type of photo ID to others turned away for not looking enough like their photo. One voter turned away is one voter too many.”
In the early afternoon, the Association of Electoral Administrators said that no major problems had been reported, though it would likely not hear about individual voters being turned away.
Chief executive Peter Stanyon said: “Polling day appears to be running as smoothly as usual, which is testament to the months of planning and hard work from returning officers and electoral administrators running today’s elections.
“We hope the rest of the day continues along the same lines.”
The Electoral Commission, which was given £5.6 million to carry out a public awareness campaign, has tasked councils with recording how many would-be voters are turned away.
No record will be made if greeters deployed outside polling stations turn people away.