Learn Hardcore Finnish: Avain voi kadota pian.

Actually, the word in question is voi, not vai. Voi is the third-person singular of the modal verb voida. Here it expresses possibility or likelihood (“may/might”) rather than ability. So Avain voi kadota pian means “The key may get lost soon.”

Why is kadota in its basic (dictionary) form after voi?

After modals like voi, the main verb appears in the first infinitive (basic form). Finnish never conjugates the main verb for tense or person when a modal precedes it. Instead of, say, kadottaa or kadotan, you keep kadota.

What’s the difference between kadota and kadottaa?

kadota is intransitive (“to get lost/disappear”). The subject (here avain) is what vanishes.
kadottaa is transitive (“to lose something” or “to make something disappear”). It would need a direct object in the partitive case (kadotan avaimen = “I lose the key”).
Since the key itself “gets lost,” we use the intransitive kadota.

What case is avain, and why is it not in another case?

Avain is in the nominative singular, the default form for the subject of a clause. Because it’s the thing performing (or undergoing) the action of the verb, it stays nominative.