From memory having worked with US military early seventies (including a year at Vicenza)........
If lady was born to US military serving at the base in Vicenza then she should have a copy of the document known as Certificate of Report of Birth Abroad (form FS -240). There is also another form known as DS-`1350 , but if memory serves me right it is the former form that her Dad should have obtained from the local US consul.
If both parents were US citizens then this would have been done automatically by US base authorities. Tens of thousands of kids were born to US military in Europe since WW2 and these forms were almost automatically issued.
If the mother was Italian I still think it would have been but am less sure.
If copy of what this, which is effectively a US birth cert for kids born abroad to US citizens, has been lost then ,with proof - passport etc, you can obtain another copy from the State dept . US Embassy should help with address and application process-and indeed everything about getting a new FS-240.
Mind you , this is all nonsense -the Register Office here should accept an E.U. birth cert and it's crazy they don't.
Actually, can the lady easily get an Italian passport ? Might make things easier...