No you won't need it, you just tell them you don't have one.

Later, if you call the provider to get a change on your account, they will request it however. It may be smart to "build" your own code on the site you sent me.

Put that in the fields:

COGNOME (Your last name)
NOME (Your first name)
COMUNE DI NASCITA : Stati Uniti
PROVINCIA : EE
DATA DI NASCITA : (Your date of birth)

And press on calcola. You will get your own Italian Social Security Number