When we did the Great Divide Trail, we asked Mark at the General Store whether we could leave our truck in town for two months while we hiked. (He’s an old friend.) He said it wouldn’t be a problem. It ended up parked in the square right in the middle of town, across from the general store the whole time. I don’t know whether Mark told the local authorities about the vehicle or not - but it was fine when we got back. Alternately, the Park may allow you to leave your vehicle at the visitors center at St. Mary’s or Two Medicine. There is also a campground in East Glacier (last year that was the only laundromat in town) that might be willing to let you park your truck. Frankly though, the center of town is busier, so the vehicle might be safer there. East Glacier is a very small town but a friendly one.
Ginny