[3.14] gh-141004: Add a CI job ensuring that new C APIs include documentation (GH-142102) (GH-142247)

* gh-141004: Add a CI job ensuring that new C APIs include documentation (GH-142102)

Co-authored-by: Hugo van Kemenade <1324225+hugovk@users.noreply.github.com>
(cherry picked from commit 1a7824a927f0706300af7bfc182884a43e2f587a)

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Co-authored-by: Hugo van Kemenade <1324225+hugovk@users.noreply.github.com>