The commissioner, Anne Longfield, wants the register to be introduced "without delay", after she found high numbers with special educational needs and disabilities (SEND) were among those who are electively home educated, and their parents struggling to cope with delivering their education.
Up to 60,000 children are known to have been home educated in England last year, according to Longfield's report Skipping School: Invisible Children - How Children Disappear from England's Schools. The report suggests that illegal pupil "off-rolling" is behind the surge.
Off-rolling refers to the practice used by some schools who seek to remove challenging or poorly performing students from the roll through tactics such as encouraging parents to take up home schooling.
The report states: "Parents who are home educating their children should be required to register their children with the local authority.
"In a survey of local authorities in Autumn 2018, all 92 respondents agreed that a mandatory register would aid them in their work."
It adds that the register should include the address at which they are being educated and parents should be asked why they are home educating their child.
Ofsted has previously suggested there are 300 schools where this could be an issue, but the watchdog's chief inspector, Amanda Spielman, said only two had been identified following inspections.
Longfield has also raised concerns that children had been withdrawn to attend illegal schools, where education and welfare standards are unchecked.
Working alongside Channel 4's Dispatches programme, Longfield found 22 per cent of the children withdrawn from school to be home educated in 2017/18 were diagnosed with SEND.
In these cases, parents recounted how they had been forced to remove their child, because the school was unable to meet their needs.
Longfield warns: "These kids can reach crisis point and without additional care from schools or from external agencies such as child and adolescent mental health services, the children fall through the gaps."
One mother told Longfield that she chose to home educate her 12-year-old son, who has attention deficit hyperactivity disorder, after he was bullied at school and temporarily excluded for fighting.
But she expressed "huge doubts" in her ability to educate him to the same level, as a result of her own dyslexia.
Funding cuts for SEND have hit schools hard, with the Local Government Association warning in December that councils faced a funding deficit of £472m for SEND services in 2018/19, and up to £800m in 2019/20.
In some cases of home schooling investigated by the commissioner, parents told of how they signed up to home education to avoid their child receiving a formal exclusion from a school, without understanding what it involved.
Nine out of 10 of the local authorities who spoke to Dispatches said they were worried about off-rolling.
A data sample from nine local authorities suggested that the number of children going into home education rose by 48 per cent between 2015/16 and 2017/18.
In some areas the increase was much steeper - 94 per cent in the London Borough of Hackney and 176 per cent in the London Borough of Newham.
Despite this, local authorities have no legal duty to monitor home educators and have no powers to visit the home to carry out checks on the education unless they have a welfare concern.
Longfield states: "Our investigations have revealed thousands of children are ‘off the grid' because they are being home schooled.
"The numbers are rocketing and no one knows how they are doing academically or even if they're safe.
"Many are being off-rolled. It also seems that a relatively small number of schools may be responsible for this sharp rise in children leaving school for ‘home education' in this way.
"Many parents who make a philosophical decision to home educate provide their children with a high-quality education.
"But there are many other families who have ended up home educating for other reasons, and are struggling to cope.
"Many of these children are very vulnerable, have SEN, or are unable to cope with a ‘one size fits all' school system. Schools should be for all children, including those with complex needs and those who struggle academically.
"We need to know who these children are, where they are, whether they are safe and if they are getting the education they need to succeed in life.
ADCS Educational Achievement Policy Committee chair Gail Tolley agreed there was a need for strengthened legislation.
"It is not good enough that we have no way of knowing whether an increasing number of children are receiving a suitable education at home or not or whether they are safe," she said.
"We need bold, decisive action from the Department for Education (DfE) that deals with the causes not just the symptoms of these issues if we are to prevent children falling through the gaps in the education system."
A DfE spokesman said only a "very small minority" of children being home schooled were not receiving the standard of education they should be.
"Last year we ran a call for evidence on proposals to introduce a register, as well monitoring of provision and support for home educators," he said.
"We will respond to that in due course."