It featured 3,500 signatures collected by Scott Fewster, whose son died of meningitis.
The charity RAPPID (Raising Awareness of Paediatric Pneumococcal Infection and Disease) also published a report predicting that universal vaccination would prevent at least 29 child deaths and more than 7,000 serious cases of pneumococcal diseases, which include meningitis and pneumonia.
Government policy is to vaccinate infants and children up to two with specific risk factors, but most diseases occur in otherwise healthy children.
A Department of Health spokesman said it was researching the issue and early results were "encouraging". He added: "When the results are complete, we will seek advice from our independent experts about whether we should introduce the vaccine, the number of doses that should be offered and the appropriate age to vaccinate children."
