Warning: People may find the photos below extremely graphic.
At the time of writing, more than 667,000 people have signed the parliamentary petition, making it the most popular since their introduction.
Support for the petition accelerated after a couple, from Maidstone, released graphic images of their two-year-old daughter Faye Burdett, who died from the virus on Valentine's Day.
Any parliamentary petition that receives over 100,000 signatures will be considered for debate in Parliament. The Government responds to all petitions that get more than 10,000 signatures.
Previously the petition with the highest number of signatures was a call to block Donald Trump from UK entry, which has 579,468 signatures.
Faye's parents, Jenny and Neil Burdett, have said the response to the petition since their toddler contracted meningitis B and died has been "overwhelming".
Dawson said he felt "absolutely helpless" as his son Sam lay hooked up to machines in hospital, and he is glad his son is now well and back home with his family.
Lee Booth, who lives in Newcastle-upon-Tyne, launched the petition in September 2015 after one of his two daughters was deemed too old to have the vaccine on the NHS.
He said he was "speechless and overwhelmed" by the "phenomenal" response to his petition, according to PA.
A vaccine to protect against meningitis B is available on the NHS for babies aged two months, followed by a second dose at four months and a booster at 12 months.
Parents who wish to have older children vaccinated must pay privately, although a worldwide shortage of the vaccine Bexsero means stocks are very low.
Manufacturer GlaxoSmithKline (GSK) hopes to have increased stocks in the UK by the summer. The NHS programme is unaffected.