The same group is also calling on drug companies to help speed the supply of vaccines by making some shots available by mid-September instead of mid-October.
But, will people actually agree to get the shot? There seems to be just as much anxiety over the vaccine as there is about the virus.
Health officials say there have been no serious problems among adults and children who started testing the H1N1 vaccine earlier this month. And, while it appears to be safe, doctors won't know until next month if it is effective. But the question remains whether most parents will even opt to have their children vaccinated.
"I don't think there is any indication at this point to believe that the vaccine poses any risk other than other flu vaccines have posed, and to put it another way, when that vaccine comes out I'm gonna be first in line," said Ken Alexander, a pediatric infectious disease specialist at the University of Chicago.
Alexander says there is more risk in getting the flu than the vaccine.
But parents are still worried about the safety of a shot that had to be created in a hurry. Alexander says they didn't really have to start from scratch.
"In other words, the technology we are using to make the swine flu vaccine is the same technology we have been used for he past 60 years," Alexander said. "Vaccines are a little like wines. You know, they are never the same from year to year, they each got a vintage, but we always make wines the same way."
Another concern: Why would we need more than one dose for this flu? Researchers say, because the H1N1 strain flu is so new, most immune systems will need extra help.
"The first dose says to your immune system, 'Wake up, pay attention, and the second dose a month later says to your immune system, 'Remember, this it's important.' "
And, like the seasonal flu vaccine, doctors say there are misconceptions they will always be up against.
"How many times have you heard the statement: I don't want to get the flu vaccine because the last time I got the flu shot it made me sick...There's nothing live in the flu vaccine. The flu vaccine can't give you the flu any more than Napoleon can pick your pocket."
Critics maintain that people getting this vaccine are still taking a risk because they will be given a drug that has not undergone a full spectrum of clinical tests.