If one is not able to make it to a minyan one day during sukkos, can he still say or do any of the day's Hoshanas services? Would it be different on Hoshanah Rabbah because of the additions?
I can't see any reason why lack of a minyan invalidates regular hoshanot.
The ben ish chai (last subsection in first year parshat haazinu)says that the main requirement according to the kabbalists is to encircle the tevah /Bimah one on each day of chol hamoed an 7 on hoshaanah rabbah ( even though no Sefer Torah is present) he mentions that people who would do it around a chair at home (who couldn't get to synagogue)are not fulfilling the true essence of the custom, "eino klum (but he doesn't say that its bad too, he adds none the less, an infirmed person who has no option of getting to a synagogue with a Bimah doesnt lose by putting a tanach on a chair and doing it at home, and quotes the verse " g-d doesn't withhold good from those that walk in his ways innocently, tehilim 84:12
On hoshaanah rabbah the custom of the Sfardim is to add some excerpts from selichot, some of which are in Aramaic "rachamana idkar lan ..." Which atleast during the selichot the custom is not to say alone " see kitzur shulhan Aruch 128:9" perhaps that would also require a minyan,however Rav Mordechai Eliyahu ztzl in his notes there brings the kaf hachaim 591:26 who is lenient on the Aramaic passages. Ashkenazim however do not seem to say those passages at all during the hoshaanah rabbah additions. The Yemenite and spanish and potugese custom is to say the 13 middot of rachamim on hoshaanah-rabbah which are not to be saide without a minyan in the form of prayer, (see previously quoted kitzur)but according to some authorities can be saide with the normal tune of reading see ben ish chai ki-tissah:9 first year.