Sara Burnett, a spokeswoman for Le Bonheur, said that while hospital workers have begun to receive vaccine, they have also received inquiries from people who cannot yet get vaccinated but want to receive it.
"We've had questions about when is it available," she said. "We've referred those back to the Tennessee Department of Health."
And some primary care doctors, responding to an ABC News inquiry, said they have not yet received the vaccine, but they have gotten plenty of inquiries.
"I'm getting pounded," said Dr. Randy Wexler, assistant professor of clinical family medicine at Ohio State University Medical Center. "People want to know when they can get it, and they want to be placed at the top' of the waiting list."
A similar rush is taking place in Washington, D.C.
"Yes, patients are asking when the H1N1 flu vaccine will be available," said Geeta Nayyar, an assistant clinical professor of medicine at George Washington University. "So much so, that they are trying to coordinate their office visits with when the vaccine is available, so they can make one office visit."
But while many people may be asking for vaccinations now, Tennessee's Cooper said she has chosen to look on the bright side.
"This is just the beginning," she said. "So again, we will continue to see vaccine being delivered to our state and hopefully within a very few weeks there will be enough vaccine for any Tennessean who wants it to get it."
Doctors nationwide are hoping the same thing.