Most of the APIs publicly available from U.S. federal agencies are read-only APIs of Open Data -- are there example of read-write APIs?
The Regulations.gov API answer is a good example! It's also an example of government dogfooding its own APIs.
As I am sure you are aware, the We the People petition site has a write API as well: http://www.whitehouse.gov/blog/2014/10/23/new-we-people-write-api-and-what-it-means-you
I have no idea if it's a good example, but there are some APIs for giving comments on regulations. See
Most sites don't allow submissions that would then be displayed immediately to other people -- it opens up too many issues including security and harrassment.
I haven't done a recent look at this in federal government, but I am slowly generating APIs.json and Swagger definitions for federal APIs. Then I will be able to keep a regular tally of POST, PUT, DELETE across federal gov APIs, not just WRITE - http://federal-government.apievangelist.com/stack.html
18F has the most comprehensive list so far - http://18f.github.io/API-All-the-X/pages/write_apis-notes