Learn Russian: Они покупают много мёда и шоколада.

This is because of the word много (a lot). In Russian, words that measure quantity like сколько (how much/many), много (a lot), and мало (few/little) force the noun that follows them into the Genitive case. Since мёд and шоколад are masculine nouns ending in a consonant, they take the Genitive ending -а.

Usually, the thing being bought is in the Accusative case. Why isn't that happening here?

You are completely right that the direct object of a verb like покупают (they buy) normally takes the Accusative case! However, the quantity word много overrides the verb. The verb is technically acting on the word много, and then много forces the actual food items into the Genitive case.

Does много change the case of both nouns connected by и (and)?

Yes, it does. Because they are buying 'a lot' of both items, the quantity applies to both. That is why both мёд and шоколад must be put into the Genitive case (мёда and шоколада).

Why does the verb покупают end in -ют?