There are two possible problems in such a case:
- The Lulav might not belong to him, and
- In a case where it's lost, the thief might not have given up on finding it, and
- This might be a case of מצוה הבאה בעבירה
1 . A stolen item belongs to the original owner, until there is יאוש בעלים (the owners give up on getting their object back), and שינוי רשות (a changing of ownership from the thief [other changes might also work, but this is the applicable one]), as per Shu"A Cho"M 396:3 (the יאוש should be before the שינוי; in a case where the latter is first, the Shu"A says he thinks that it still belongs to the original owner). Therefore, assuming this is a case where the person who was stolen from gave up on finding his lulav, the lulav now belongs to the buyer, because there was 'giving up' and 'a change in ownership'. If the owner didn't give up, then the lulav doesn't belong to the buyer.
In the second case given, where the lulav was found, this leads us in to the realm of השבת אבידה, which is lengthy. In short, the owner must give up on finding his object. Therefore, if it has signs which the owner can readily identify his object by, we assume he didn't give up and the lulav doesn't belong to the finder. If it has no signs, and the owner will notice its loss, we assume he gave up on finding it. In our case, though, we might not have to worry about this at all, seeing as the thief doesn't actually own the object, and we're under the assumption (from the previous point) that the original owner already has יאוש.
מצוה הבאה בעבירה - doing a mitzva while simultaneously committing an Aveira, is complicated. Here, the buyer is not stealing. Beyond that, he's annus and didn't know the lulav was stolen. The Aveira he did commit is מחזיק ידי עוברי עבירה (supporting another's sinning), which is a form of Lifnei Iver (Shu"A Cho"M 396:1). However, since it could be committed by others, it's not fully Lifnei Iver and it's only d'Rabanan (אשל אברהם סעיף קטן ב, among others).
The S'dei Chemed (chelek Dalet, klal 77) talks about many aspects of mitzva haba'a b'aveira, and brings a machlokes by a case where the Aveira is d'Rabanan, and a machlokes where the Aveira is done b'oneis. In the second machlokes, he holds that there's still mitzva haba'a b'aveira. In the first, he leans towards the side that it's muttar.
I didn't see anyone (quoted by the S'dei Chemed) who holds that in both cases (annus and d'Rabanan) there's still mitzva haba'a b'aveira, but there are definitely Achronim who are machmir in either of the two cases, so it seems to me that it'd be preferable to shake lulav again. However, if that's unfeasible, seemingly יש על מי לסמוך, and he's yotzei. And in a case where he found the lulav, there's no problem of מחזיק ידי עוברי עבירה at all,and assuming the previously mentioned problems are dealt with, he would be yotzei with this lulav.