As far as I'm aware, as a native citizen of the USA, there is no term for this.
Generally, if we need something from a government agency, there's a form we fill out and submit (likely with some required documentation). That's it. We don't necessarily need to write a letter to go along with it and, if we do, that letter doesn't have a specific name.
For example, when I recently needed a copy of my son's birth certificate, I found the form on the city's website, filled it out, and then had my husband turn it in at their office in person. I could have submitted it digitally or by mail but that would have taken longer (and cost more). If I'd mailed it, I wouldn't have enclosed a letter. The website linked above details the necessary enclosures:
- Effective immediately, all mail-in requests require notarized proof of identification.
- A completed application (download and print one in English or Spanish)
- A check or money order payable to the Office of Vital Records ($23 for each copy of a birth certificate; $21 for each death certificate plus $4 for each additional copy)
- A copy of your current government-issued photo identification
A copy of proof of residence, such as utility bill, voter registration, etc., if different than the address on your photo ID
- A copy of a current government-issued identification of the person who signs the check (if different from the requestor)
- A stamped self-addressed envelope (use an express mail envelope for express mail delivery).
None of the things listed above include any sort of letter, though the entire submission is referred to as a "request".
So, if you're filling out a form, that's called an "application" and if there's not a form, you're just sending a "request". There's no specific word.
