Well, what a crappy situation to be in on what is supposed to be one of the happiest days of your lives.
I think in that situation I would first contact my close friends and family by phone call and be honest about the situation of a suspected theft, without saying that you suspect anybody, and ask if they left a card or gift/cheque etc. Ask your wife to do the same with her friends & other family. Document what they tell you. Having told them this you can send a more generic thank you card and they will understand if it doesn't specifically mention their gift. For those you know less well a generic "thank you" card is probably fine anyway.
If you do clearly establish that a number of cards/gifts are missing, which it seems you almost certainly will, you then have a decision point.
Either it is a criminal theft, so you will report it to the police. Your wife might have the option of calling her mother in tears etc. saying what has happened etc. before contacting the police, in case her mother wants to own up at that point to avoid police involvement. Once the police are involved you then have to be prepared to press charges against the MIL. The police would likely be able to establish who did it e.g. by tracing the cheques and/or tracing the serial numbers of gift cards, if they are willing to investigate it that far (but they may see it as a family dispute not worth their time).
That your mother in law one day had no money and after the wedding had enough for a plane ticket has no reasonable explanation short of a lottery ticket win.
Or, as you say, you decide to drop the matter as you don't need the money and the hassle and implications on family relationships are too severe.
I think I would discuss with your wife but make it clear it is ultimately her decision as it's her mother, and you will support her decision either way. Going down the route of the police investigation is effectively saying you will both terminate the relationship with the MIL.
A person with an ongoing addiction is unfortunately likely to be a very difficult and stessful person to deal with, I would be inclined to minimize contact, meet them only in neutral places and not have them in my home, unless the person is willing to seek professional help and turn their life around.