Nothing is shown in the movie about the apes manufacturing weapons and ammunition but we’re led to believe they do craft their own guns: for example, the bulky carbines they use don’t look like any real-world rifles that I can think of.
Actually, this site says they're modified M1 carbines, which only half answers your question: this means the props are made out of real-world carbines; but in-world, modifying an existing (and working) piece of weaponry would make little sense.
So we'd rather take these weapons as an original design by the apes.
In the sequel, Beneath the Planet of the Apes, the apes use an even fancier gun: a few years have passed from the events shown in the previous movie (I can’t remember how many) and this kind of rifles could be an improvement over the previous, which might suggest they’re able to make their own weapons (the same site as above says the props are actually modified Madsen M50s).
In this movie the apes are also shown trailing and loading their own cannons, that look like 19th century muzzleloaders, which wouldn’t be available in the 20th or later Earth: as it’s an outdated technology for the 20th century Earth, it’s very likely the apes have discovered the way to produce them themselves. (Sorry, I couldn’t find any picture of these cannons online).
This seems to comply with Pierre Boulle’s original novel, whose story is set on a different planet than Earth until the final scenes: and although the apes of this faraway planet revolted to their former human masters while «wielding a parody of a weapon» (that is, whips), through the novel they are also shown self-sufficient in everything.
Hence, the reader is led to believe that even the guns they use to hunt humans are made by the apes.
In conclusion, I would say that the apes in the movie make their own weaponry.
How they reached this I can’t say: they either copied the technology from the human leftovers or developed it by themselves, which I think is more likely and in tune with the setting and the novel.
Probably they started with basic flintlocks and then over the centuries reached the technological improvements needed to make 20th-century-alike weapons. And even go beyond them.