The correct sentence is:
I think my card was stolen by my friend.
Here, you use the past of is because the action of stealing has completed. The card is gone; it is not currently becoming stolen. Here's an expanded sentence with the same meaning:
I think my card has entered and is now in the state of being stolen, which was caused by my friend.
Using the past tense does not imply in this case that the card is no longer stolen. When something is described as stolen, it is generally understood that it remains so unless otherwise noted. Unlike, say, running, the state of being stolen does not naturally terminate. So, saying I think it was stolen does not imply that the object has been recovered, while saying I think he was running does imply that he is no longer running, because we know that people stop running after a relatively short time.
The alternate sentence (is stolen) sounds very awkward because of these reasons, though I don't think it's not grammatical.
If you want to make it more explicit that you still don't know where your card is, you could say:
My card is missing, and I think my friend stole it.
If you want to convey that your card is currently the target of a theft in progress, you could say:
My card is being stolen by my friend.