Are you able to sue companies that send you preapproved purchase and/or debit cards in the mail without your permission?

I do not know about suing them but here's how you opt out of receiving these: Consumer Reporting Companies If you decide that you don't want to receive prescreened offers of credit and insurance, you have two choices: You can opt out of receiving them for five years or opt out of receiving them permanently. To opt out for five years: Call toll-free 1-888-5-OPT-OUT (1-888-567-8688) or visit www.optoutprescreen.com. The phone number and website are operated by the major consumer reporting companies. To opt out permanently: You may begin the permanent Opt-Out process online at www.optoutprescreen.com. To complete your request, you must return the signed Permanent Opt-Out Election form, which will be provided after you initiate your online request. for this and additional consumer information see Federal Trade Commission website: http://www.ftc.gov/bcp/edu/pubs/consumer/alerts/alt063.shtm

You haven't suffered any harm, so you don't have a right to sue anyone. Look more closely at what you're receiving, though. Since it is illegal under the Truth In Lending Act to send anyone a credit card unless they apply first, they are not sending you an actual card that can be used to purchase anything. They may be sending you a "loyalty" card or some other promotional thingy to try to induce you to visit them and buy things from them. I think it unlikely that someone could use one of these cards to perpetrate identity theft.