I'm wondering - what relations does the GOBPANCESTOR function in the GO.db package use to collect the ancestors of a given GO:BP class? Is it just the is-a relations, or does it include part-of and/or regulatory relations as well?

It includes all of those:

library(GO.db)
library(DBI)
con <- GO_dbconn()
table(dbGetQuery(con, "select * from go_bp_parents;")$relationship_type)

                is_a negatively_regulates              part_of 
               57509                 2813                 5388 
positively_regulates            regulates 
                2796                 3240