Ancient Europeans had the sulphur, just no rubber trees.
Ancient Mesoamericans vulcanized rubber (e.g. to make the ōlli for Ollama) by extracting latex from the Panama rubber tree and mixing it with moonflower juice:
Those balls (which were solid rubber, very heavy, and deadly) are the only Aztec use of rubber we ever hear about. The other uses mentioned are more interesting:
> they identified a number of ancient hollow rubber figurines, a band made from rubber that secured a stone axe head to its wooden handle and numerous small rubber balls. The 16th-century documents also mentioned that Mesoamerican people made rubber-soled sandals and rubber-tipped hammers and drumsticks.
I mean, phrases like "ancient hollow rubber figurines" and "a stone axe head secured by a rubber band" are delightfully incongruous.
Ancient Mesoamericans vulcanized rubber (e.g. to make the ōlli for Ollama) by extracting latex from the Panama rubber tree and mixing it with moonflower juice:
https://news.mit.edu/1999/rubber-0714