I would like to know if there are any authors from the period of the rishonim or earlier who have compiled a running commentary on birkat hamazon.
Rabbi Zedekiah ben Rabbi Avraham HaRofe wrote a short commentary in Shibbolei HaLeqeṭ 157.
Seder Rav Amram Gaon (starting at page 350 =175b of the printed page numbers) has a running commentary.
Rabbi Eleazar of Worms has a commentary in siman 147 of his commentary on the Siddur (פירושי סידור התפילה לרוקח).
(h/t to @Matt for some of these)