The IRS form 1040, Schedule A and form 8283 all provide the space for tax return filers to list their SS# and I don’t believe that filers have the option or the right to leave it blank.
But if you itemize, you just fill in the total amount of your contributions in Schedule A, broken down into “Cash or check” and “other.” The IRS doesn’t ask for a list of the organizations unless you’re audited.