ZB, I think that sounds nice. I would give her the card. I believe I did something similar with my in-law's (including brother and sister-in-laws) and said something to like I was looking forward to spending time with them or getting to know them better, or something like that. They all thought it was sweet. They all gave me "Welcome to your new house" cards when I moved in, which was really touching. Plus housewarming gifts that included things like a brand new washing machine and other astonishing generosities.
I can relate to having your in-laws be pretty upset about a fast marriage when they don't know you very well. I spent an evening with DH's parents and we all got along great (we stayed up drinking Metaxa brandy until 2am, it was a riot!) but they still got upset when DH announced our engagement just after I met them and that we were getting married 2 months later in America. It was a big blow for them. His mother lost a lot of weight because she was really upset about it all (mostly upset about how DH's baby mama was going to handle it - I think MIL was really worried about the ex freaking out and not letting anyone see the baby anymore, which was understandable.) And I think they were also worried that I would back out of moving here, or move here and then leave a few months later because I hated it or something, and leave their youngest son broken-hearted. I can see why they were worried. They did make an effort to show me around town and include me in things when I first moved here, though there was a bit of confusion between us because of our accents. It was a bit awkward at first, and my MIL is sort of rigid when you first meet her - not like my family members, who just dive in for a hug for the first time they meet someone! But I have realised what a warm and caring person she is underneath the tough exterior and I know she truly loves me and cares about me now. Don't worry, it will just take time.