Learn Swedish: Jag har ingen aning om vad hennes adress är.

Because vad hennes adress är is an indirect question, which functions as a subordinate clause (a smaller sentence inside the main one). In a direct question, the verb comes right after the question word: Vad är hennes adress? But in an indirect question, the subject (hennes adress) comes before the verb (är).

Why do we need the word om here?

The Swedish expression for having no idea about something is att ha ingen aning om. Even when it is followed by a question word like vad, Swedish keeps the preposition om. In English, you usually drop the word about and just say no idea what, but Swedish keeps it.

Why do we use ingen instead of inget?

The word for idea or clue, aning, is an en-word (en aning). Therefore, we must use the matching en-word form of no, which is ingen.

Could I just use vet inte instead of this longer phrase?

Yes, you can absolutely say Jag vet inte vad hennes adress är. Using Jag har ingen aning om is just a stronger way to express it, exactly like choosing I have no idea instead of I do not know in English.