Varies by band. Some, like Blues Traveller, have a lead singer that play an instrument, so they're involved in the jam. Others grab a background instrument like a small percussion kit, tambourine or shaker and get involved in creating the groove. Others are merely an extension of the audience and dance and move around, and still others simply walk ...
As far as I know the jam band scene (I'm a big Umphrey's McGee fan) do bands like them (Phish, String Cheese Incident, Moe, Disco Biscuits etc.) don't have members that only do vocals. In fact their focus is on playing their instrument, that is where they shine and imho they are far better at that than singing. Sometimes I wish UM had a dedicated singer.
An unexpected album of theirs, which non-Deadheads like me often like is Workingman's Dead. The material is more folk/country/Americana. The review here gives an idea of how unexpected it was, and how it also propelled them to recognition and success... Other details in wikipedia.
Arguably, the "jam band" format is a throwback to an older style of music found in many cultures. I'm not an expert, but I believe extended, semi-improvisational variations on a central "groove" is more the rule than the exception for traditional music in Africa, India and the Americas, as well as among the Celtic parts of the British Isles.