Italian media report that regional authorities are handling the situation in a number of different ways.
In Bologna, the local authority has set letters of suspension to the parents of some 300 children, and a total of 5,000 children do not have their vaccine documentation up to date.
In other areas, there have been no reported cases, while still others have been given a grace period of a few days beyond the deadline.
Is the law having an effect?
The new law was passed to raise Italy’s plummeting vaccination rates from below 80% to the World Health Organisation’s 95% target.
On Monday – the last day for parents to provide documentation proving their children had been properly vaccinated – the Italian health authority released figures claiming a national immunisation rate at or very close to 95% for children born in 2015, depending on which vaccine was being discussed.
The 95% threshold is the point at which “herd immunity” kicks in – when enough of the population is vaccinated for the spread of the disease to become unlikely, thereby protecting those who cannot be vaccinated.
That includes babies too young to be vaccinated themselves, or those with medical conditions such as a compromised immune system.
Last month, an eight year old recovering from cancer was unable to attend school in Rome due to his weak immune system.
The child had spent months receiving treatment for leukaemia, but was at risk of infection because a proportion of pupils in the school had not been vaccinated – including several in the same class.
Demonstrations against compulsory vaccination were held in Rome, 2017
The Lorenzin law, drafted by the previous government, had a tumultuous birth. When the current coalition came to power, it said it would drop mandatory immunisations although it later reversed its position.
The two populist parties in power had faced accusations that they were pursuing anti-vaccination policies.
Writing in a Facebook post on Monday, Ms Grillo admitted it “is a law that, at the time of approval, we criticised for several reasons” – and said that the law would be changed to include only those vaccinations that were necessary based on scientific data.
Why do parents not immunise their children?
The anti-vaccination movement has been growing globally in recent years, sparking alarm from the World Health Organization.
A long-discredited paper by Andrew Wakefield was behind much of the scare but rumours around immunisation have continued to spread, leading to public health risks as not enough people are immune to such diseases.
Mr Wakefield was struck off the UK medical register after fraudulently claiming there was a link between the measles, mumps and rubella vaccine (MMR) and autism and bowel disease in children.
He made the claim based on the experiences of just 12 children, and no other study since has been able to replicate his results.