Oh dear, rookie mistake number one, today!
It's been a long time since I've done any machine sewing; well over ten years, in fact. So I completely missed the fact that there was a shuttle and a bobbin involved, despite the big clue from this being a 'vibrating shuttle machine'. But thanks to this amazing youtube channel, which I've also linked to in my resources post, I opened the sliding plate on the machine and found this amazing piece of metalwork. The shuttle comes out really easily by pressing on the point with a finger, but has to be clicked back into place, a point that worried me for a few minutes when I thought it wouldn't go in!
After another youtube video, I was able to wind the bobbin (turning the bobbin winder by hand, unfortunately; the wheel on my Singer is so well-used that it's oval, and it's the one thing I forgot to get a picture of) and load the shuttle with my purple thread, pretty easily after the first try. As proof that it's done correctly, this was taken while the shuttle was hanging by its thread from my hand, so it's all properly in place.