First, have you talked to your DH about this? Does he notice it (chances are he doesn't, b/c he probably grew up with it)? Knowing his mother, can he give you any insight as to how to handle this, or if she's just making small talk or really IS fishing for more thanks?
Second, what kind of a thank you do you give? Are you a quiet person and is MIL more boisterous? I ask b/c perhaps she doesn't take your polite thank you as sincere or something, so she keeps asking so she can ascertain whether or not you really DO appreciate it. I'm just wondering if you give a proper, polite thank you, and she expects just a bit more gushing or just conversation about the deed or the item at the time (like about the baby clothes). I know with my SIL, when we give her and her family presents, I'm often left feeling kind of used or just unappreciated, b/c they'll often throw a haphazard 'thanks for everything' over their shoulder as they walk out the door. No appreciation for the individual item, or genuine indication that she/they appreciate the gifts at all. So if MIL is trying to do something nice and is expecting .... more, and she doesn't get it, she feels compelled to bring it up b/c she wants more validation that her gesture is truly appreciated. Face it, even if you don't like something, polite people will still say 'thank you'. So she may want more validation that you're not just being polite, and that she really is being helpful. Or maybe, like other PPs have suggested, try to bring up the item in conversation before she does, not as a thank you, but just to bring it up to let her know that you still remember that she bought the item/did the favor for you. That may be enough validation for her without having to tote out another obligatory-sounding 'thank you'. Like "Hi, MIL. I was just doing the laundry. I washed some of those clothes that you bought for baby. I can't believe how little they look!"
Please don't take this as trying to blame you, because I'm not. People communicate differently and have different expectations, and this can be cause for hurt feelings and resentments to pop up. This is just coming from someone who has a MIL who communicates a bit differently than me. It was always a sore spot, but now, with age (LOL), I've come to accept it and roll with it and appreciate the positive points, rather than let the negative points annoy me so much. Sometimes I think that now, I talk to her even more than my DH does!
Third, if you try the conversation thing and it seems like she's still fishing for the actual words "Thank you", then perhaps you could nicely point out that you DID already thank her for the favor/item, without really putting her on the spot? Maybe (if you're a decent actress), try to go embarrassed-sounding. "Oh, I'm so sorry, MIL....did I forget to say 'Thank you' for the clothes? I really thought I did at the time...I always make it a point to thank people when they do something nice for me. Well, if I forgot, I do apologize. Thank you." Look her right in the eye when you say the last thank you. Then let a beat or two go by and change the subject.
I want to issue the disclaimer that this is for things that are truly helpful, like perhaps buying the baby clothes (it does save you money to buy other things, even though you know you can afford it). If she wants to do something that you really don't need her to do, at all, like walk the dog again, then simply refuse. You don't have to provide her with every single way that she can think of to prove her usefulness. "No thanks, MIL, I just walked him. No, really, he doesn't need another walk. No, but thanks anyway. I'll let you know if I ever need someone to walk him."
ETA: Just saw your update. Well, then, I would go with outright refusal of favors (politely, of course): "No, thank you. I can handle it" with 'it' being walking the dog or buying baby clothes. And if you do let them do something to save the peace, then try the polite way to call them out: "Oh, I thought I already said thank you when you bought the clothes. Didn't I say 'thank you'?"
And maybe you should consider limiting their access to you, your lives, and your home a bit more. I mean, how do they even have access to your DH's email? Lock the computer or the phone!