It looks a lot like a big old round meteor crater, but isn’t listed as one on the sites that keep track of such things. Is that just because no one has checked, or is there some other theory which explains the Bays origin and peculiar shape ?
Accordring to this site, Hudson Bay is underlain by 1800m (that’s more than a mile) of sediment dating back to the Paleozoic (at least 300 million years). The basic form (with thickest sediments in the center of the basin) suggests sedimentation within an existing structure, but I have been unable to find out what this structure is. Chances are it’s just fault bounded basin that has slowly been rounded out recently (like the last 100 million years or so).
Hudson Bay I don’t know about (though it doesn’t look all that round to me), but the arc just above James Bay probably is a crater. I couldn’t find a link on the brief search I did, but remember a map showing concentric ridges around it and a description of shattered rock that suggests an impact.