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.
1Just about every author of a peirush on the siddur included a commentary on birkas hamazon. This includes the Ri Ben Yakar, Abudarham, and Machzor Vitri– הנער הזהDec 21, 2015 at 1:15
1Also the Shibbolei Haleket 157 and Kol-Bo 25 discuss several of the phrases in birkas hamazon, commenting on them– הנער הזהDec 21, 2015 at 1:17
The Abudarham has a running commentary here (in Sefaria, it starts at paragraph 46ff).
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).
You can also look at the Tur on Orach Ḥayyim simanim 187-189.
(h/t to @Matt for some of these)