Aleks pointed me to this site by Alexander Genkin, David D. Lewis, and David Madigan that has a program for Bayesian logistic regression. It appears to allow some hierarchical modeling and can fit very large datasets. I haven’t tried it out yet but it looks pretty cool. I imagine that for some complicated problems (for example, estimating state-by-state time series of public opinion), it probably wouldn’t work “straight out of the box”–but that’s fine, nothing else is available to solve these problems. The good news is that the program of Genkin, Lewis, and Madigan is open-source and (apparently) fast, so it could be possible and worth it to go inside and adapt its code as necessary to fit more complicated multilevel models.
P.S. Here’s the paper. According to Yu-Sung, they use a one-variable-at-a-time update, so maybe some rotation would speed things up.