Hmmm. There used to be a ton of vinyl shops when I lived there, but that was fifteen years ago. You may want to check whether Nuggets (in Kenmore Sq.) is still around, or Planet (which used to be in Kenmore, but the entire block where it used to be was razed for a luxury hotel, so they're now out in Harvard Sq, just down JFK St., I think). Mass Ave between Central and Harvard Squares used to be filled with shops, though as a previous poster has indicated that may no longer be the case. One of the last times I was in Boston, a few years ago, there was a decent place on Mass Ave further out, maybe almost all the way to Davis Sq., though I can't remember the name or exact location. Sorry! Do check out In Your Ear on Mt Auburn St when you're in Harvard Sq., as well as their shop out by the BU arena, which I assume is the shop Hammergjh mentions--IYE used to have a shop there, anyway. They shut down their Providence branch two years ago, and told me at the time they were planning to bring a bunch of vinyl up to Boston. Finally, years ago there was a Looney Tunes branch just off Mass Ave near Berklee College, too, and, at least a few years back, a rather overpriced place just around the corner on the west end of Newbury St.
Hope this helps.