Actually we know this. It was the Russians, specifically the KGB, I believe. As I recall it was because of his support for the Solidarity movement in Poland. I could be wrong, though, so if some other FReepers remember it better, please correct. I do know it was the Russians, though.
There's no doubt it was the Russians behind JPII's shooting.
I've always thought it more than a coincidence that he and Reagan were shot within months given the project they were working on together.