I didn't do this lesson, I tested into level 8 so I'm just reviewing "weak words". 時計を買います is a completely acceptable translation for this sentence. Maybe have "wrist watch" in the English sentence if that's what you're looking for.
Both とけい and うでとけい should be accepted translations for "watch". If anyone sees とけい marked wrong as a translation for "watch", please report it.
From other exercises I noticed they translate とけい to Clock and うでどけい to Watch...
Yes, this is confusing. I took a year of Japanese at university, and my native Japanese sensei taught us とけい as watch. As a general word for the item, not necessarily a wrist watch..
とけい is correct for both watch and clock in a general sense