Your thought is absolutely correct, and it is indeed Akkusativ of the possessiv pronoun (which incidentially is identical to the nominative case for these). Your question is far from stupid.
The base sentence without saying whom it belongs it is (same problem: Akkusativ looks like Nominativ here):
Sie kaufen das Fahrrad
Kaufen can take two objects, where the Akkusativ always refers to the object being bought, and the Dativ can refer to a person whom the thing is bought (jmd. etw. kaufen): "Ich kaufe Dir sein Fahrrad"
Ich: Subject, Nominativ
kaufe: verb
Dir: Dativ, whom (Wem?)
sein Fahhrad: Akkusativ, what (Wen/Was)? where 'sein' is a possessive pronoun replacing the article; it follows always the case of the object it belongs to. E.g. similarily:
Ich kaufe Deinem Hund ein Leckerli (= I buy your dog a treat)
"Deinem Hund" is the Dativ object whom something is bought, but the possessive pronoun "Dein" thus has to be in Dativ, too, thus "Deinem".
So, back to the sentence at hand: If you now want to describe whose bike is being bought, you can replace the article by a possessive pronoun which has to match the noun in numerus and genus (thus declension). It is like in singular (Nominativ - Genitiv - Dativ - Akkusativ):
mein - meines - meinem - mein
And in plural:
meine - meiner - meinen - meine
As a side remark: you might want to skip thinking of direct and indirect objects; it's a concept in English, but not in German. Thus it does not (always) translate well and is a 90% rule at best. You have to learn what verb takes which kind of objects for what purpose.