I think there are two separate things that could be at play here:
1. as a PPer stated, the founders attended church together. I know that many years ago, there was a big bruhaha about whether or not LLL should be a single-purpose organization. There were a number of people (including founders) who wanted LLL to be officially pro-life/anti-abortion. That proposal was rejected, and LLL remains a single-purpose organization. They don't take an official stance on issues unrelated to breastfeeding.
2. Many groups meet at churches. One of the groups that I led met at churches; during the time I was involved in the church our group met at 3 different churches. We investigated many other meeting place options, including community centers, schools, libraries, hospitals, a health food store, etc. Churches were the only place that offered free meeting areas for us. So while we weren't officially associated with any of the churches, it is often one of the only options for LLL groups, especially for groups that are too large to meet in homes.