What is a typical way of saying "Here are some apples"? As in discovering them, or offering them to someone? Is it this sentence with different word positions, or different altogether
If you offer them to someone, you can say:
"Itt van néhány alma".
You can also add "tessék", meaning something like "here you are":
"Tessék, itt van néhány alma".
If you are discovering the apples, both "itt van" and "van itt" are fine, emphasizing "itt" in both cases.
Why isn't it 'vannak' here? because these ARE several apples (although the Hungarian Language marks it as alma instead of almák).