The Privacy Act is triggered by "the collection, maintenance, use, and dissemination of personal information by federal executive branch agencies" - not whether a form is approved by the government or used in a transaction between an individual and a private business.
Here is a list of Federal Branch Agencies, filling out paperwork for any one of these branches would be protected under the Privacy Act.
Executive branch
3.1 Executive Office of the President
3.2 United States Department of Agriculture
3.3 United States Department of Commerce
3.4 United States Department of Defense
3.5 United States Department of Education
3.6 United States Department of Energy
3.7 United States Department of Health and Human Services
3.8 United States Department of Homeland Security
3.9 United States Department of Housing and Urban Development
3.10 United States Department of the Interior
3.11 United States Department of Justice
3.12 United States Department of Labor
3.13 United States Department of State
3.14 United States Department of Transportation
3.15 United States Department of the Treasury
3.16 United States Department of Veterans Affairs
Financial and medical paperwork notwithstanding, who else would you fill out government originated forms for which contain sensitive information?
This is from the link gc70 provided
A. The "No Disclosure Without Consent" Rule
“No agency shall disclose any record which is contained in a system of records by any means of communication to any person, or to another agency, except pursuant to a written request by, or with the prior written consent of, the individual to whom the record pertains [subject to 12 exceptions].” 5 U.S.C. § 552a(b).
It is this rule coupled with the fact the just about any government form you fill out would be under the jurisdiction of one of these agencies. Therefore, it stands to reason that because the number of agencies is so wide and varied and that they are all covered under this rule that any government form filled out would be protected as well. Unless it was made clear beforehand that said form was not covered.