WordNet records no synonym and no antonym for barnstorm. It is the only member of each of its synsets, which is common for technical, taxonomic and single-sense vocabulary — there is no other English word that means the same thing. The broader and narrower terms below are the useful relations in that case.
Counts are distinct word forms across every sense. A synonym that appears under three senses is counted once here and shown three times below, under each sense it belongs to.
appear at county fairs and carnivals as a stunt flier and parachute jumper
tour the country making political speeches, giving lectures, or presenting plays
“The presidential candidates are busy barnstorming this month”
Synonyms, antonyms and the broader and narrower relations for barnstorm come from Princeton University’s WordNet. Full sources and licences.