If you got your shots at the Board of Health, you got a card that listed when you got them. You had a copy to keep in your wallet or with your important papers, such as your birth certificate. My son, born in 1969, had a similar card that they gave me when he had his first baby inoculations.
Now when you get flu shots, whooping cough vaccine, pneumonia, shingles shot etc, you may not get all of them in the same place. Many pharmacies offer them, as well as the board of health or your primary care physician. There should be some way with the technology we have today to be able to link these records so no matter where you get the shots/vaccines, the information is held in one record that you could access online to get an Up to date record of your shots to use for just such a thing as travel or enrollment in school. Electronic medical records ought to be able to solve the problem. Guess we will have to wait and see how this turns out. 
On edit: I agree that this will not solve the problem of us geezers who are clueless about which shots we got and even which illnesses we had. My husband does not remember having chicken pox and his Mom is no help. She can't remember if he did or didn't have that. Maybe a simple blood test could be developed or is already available to answer those questions.
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