Learn Finnish: Asiakas oli täysin kiittämätön myyjälle. - The customer was completely ungrateful to the seller.

It is driven by the adjective kiittämätön (ungrateful). In Finnish, adjectives that describe an attitude or behavior directed at someone—such as being friendly, mean, or ungrateful—typically require the target person to be in the allative case (-lle).

The verb kiittää (to thank) normally takes the partitive case. Why doesn't the target take the partitive here?

Because kiittämätön is functioning as an adjective of attitude, not an active verb. While you thank someone using the partitive (Kiitän myyjää), describing your character trait or behavior towards someone shifts the rule to the allative (kiittämätön myyjälle).

How exactly is the adjective kiittämätön formed?

It is formed from the verb kiittää (to thank) plus the negative suffix -mätön / -maton. This suffix creates an adjective meaning that someone or something doesn't do the action. Literally, it means "un-thanking".

Could I use kokonaan instead of täysin to say "completely" here?

Not really in this context. Kokonaan usually implies physical completeness (like eating a cake completely). For abstract qualities and adjectives like "ungrateful", you should use täysin or the more colloquial ihan to mean "completely" or "entirely".