Learn Finnish: Heillä on yhdeksän tai kymmenen.

Finnish doesn't have a direct verb for "to have". Instead, it uses a structure that literally means "on [someone] is". He means "they", and the ending -llä changes it to heillä ("on them"). Add the verb on ("is"), and you get "they have".

Why do we use the singular verb on if they have a large amount of things (nine or ten)?

In Finnish "to have" sentences, the verb is always the singular on (is), never the plural ovat (are). Even if they possess nine, ten, or a million things, the structure remains heillä on.

If I wanted to add a noun, like "They have nine dogs", how does the number work with the noun?

The number yhdeksän stays in its basic form, but the noun that follows it must take the partitive case. "Dog" is koira, so "nine dogs" becomes yhdeksän koiraa. This important rule applies to all numbers greater than one.

Could I use the word vai instead of tai for "or" here?

No. Tai is used in statements like this one to express alternatives. Vai is also translated as "or", but it is strictly used in questions when you are asking someone to choose between specific options (for example, "Do they have nine vai ten?").