distribute has 26 distinct synonyms spread across 6 of its 10 senses. Each sense below keeps its own list, because a synonym belongs to a meaning and not to a word: swapping one sense's substitute into another sense's sentence is how a thesaurus makes writing worse. WordNet records no antonym for it — antonymy is a relation between word forms rather than concepts, so it is deliberately sparse.
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.
distribute or disperse widely
“The invaders spread their language all over the country”
make available
“The publisher wants to distribute the book in Asia”
cause to be distributed
“This letter is being circulated among the faculty”
cause to become widely known
“spread information”
spread throughout a given area
“the function distributes the values evenly”
be distributed or spread, as in statistical analyses
“Values distribute”
Synonyms, antonyms and the broader and narrower relations for distribute come from Princeton University’s WordNet. Full sources and licences.