. This light fixture is a track to which things attach. It came with spots; I've been asked to see if I can't 'shed more light,' so I'd like to go see if I can acquire alternative things. Can anyone help me identify the manufacturer and whether they still make/sell lamps that attach?
Lamps (bulbs) you will definitely find, additional heads you will most likely not.
That stuff is cheap home center lighting. The type of thing they'll have in stock for a few months and then you may never see it again. That, or they'll get some back in from a new importer that looks similar but is functionally different. Also, parts for things like this are basically nonexistent. If you do happen to find it you only bet will likely be to buy a whole additional set and use just the heads.
Always turn off the power.
What it looks like you have is Bendable/Flexible track lighting. If you look at the junction box that the light came with and remove the canopy it may have the manufacturer name on a sticker on the inside. Now finding exact match parts may not be available, but if you have the voltage that track/lamp is, you may be able to find a a suitable substitution.
As for the lamp yes they still sale them, I would go to a flood lamp it gives a wider amount of light.