Learn Italian: La signora ha bisogno di aiuto.

In Italian, expressing a need is usually done with the phrase avere bisogno di, which literally translates to "to have need of". Here, avere (to have) is conjugated to ha because we are talking about a third person (la signora).

How would I ask this lady "Do you need help?" using the formal register?

You would say: Signora, (Lei) ha bisogno di aiuto?. The formal "you" (Lei) uses the exact same third-person verb form (ha) as when you talk about someone. Notice that when speaking directly to her, you drop the La before signora.

Why is there a La before signora?

In Italian, when you talk about someone using a title like signora (Mrs./lady) or signor (Mr.), you must put the definite article (il or la) in front of it. You only drop the article when you are speaking directly to the person.

Why does aiuto end in "-o" if we are talking about a woman?

Aiuto is a noun meaning "help" or "assistance", not a describing word (adjective). Because it is a noun, its grammatical gender is fixed as masculine. It will always be aiuto, regardless of who needs it.