With the word "to" removed, both forms are grammatical, with slightly different meanings. Both "have" and "make" in this causative use take a bare infinitive, without "to" (the
plain form of the verb):
Google Books Advanced English Grammar
Like verbs of the senses, verbs with causative meaning like make and have
(verbs that communicate that someone wants someone else to do something)
are complemented by an NP followed by a bare infinitive.
A is responsible for having B do sth.
A is responsible for making B do sth.
The second version sounds more forceful. It might suggest that B didn't want to do sth.