Is this sub panel fed with 3 conductors or 4?

You obviously meant buy a ground bar (rather than a neutral). In addition to the additional bar, you need to remove the bonding between the existing bar and the case (and obviously move the ground and neutral feeder conductors as well as the ones for the branch circuits to the right bars).

In addition to Joe's questions as to whether they properly ran a ground as well as the neutral to this panel, the rest of the answer depends on the manufacture of the panel.

How this is accomplished depends on the panel. Some panels already have multiple bars that are bonded together and you just remove the bond and use the one electrically isolated from the case as the neutral bar. In others, you install an additional bar as the ground and remove the bonding (often it's just a screw) to isolate the neutral bar from the case (and ground).

Subpanel is fed by 12/3 wire from the main. It goes into a small cooper branded (old) two breaker box, and the breakers look like Bryant style. The two hits connect to each side like in every panel I've seen, but the neutral is up top between the two hots on a plastic holder. I will check for continuity via a bonding screw though.

The only thing wrong is the bonding of neutral and ground. The neutral busbar has black plastic standoffs, so if I drilled in a neutral bar I think I'll be set.

I never saw any issues. In theory if I was running big loads, or perhaps unbalanced loads (20a on the one breaker and none in the other in the subpanel), would/should I see some issues with voltage since neutral and ground are currently bonded in two spots?
Thanks!

If it is 12/3 you have no separate equipment ground, unless you mean 12/3 with ground. Connecting the grounds and neutrals creates the potential for energizing things that are not supposed to be energized.

If you do not have a separate grounding wire in the 12/3, or a grounded conduit, DO NOT simply move the branch circuit grounding conductors to a separate bus, report back to us so we can discuss your options.

Yes, 12/3 with ground, sorry.

You keep talking about installing a new neutral bar... this is almost certainly wrong. The one connected to the case is the GROUND bar. The neutral bar must be isolated from the case.

Ar there multiple short bars that the neutrals/grounds are landed on? In that case there is likely a bond strap between the bars that can be removed (or perhaps it already has and there's nothing wrong with your pane). However, this doesn't sound like any Bryant panel I've heard of. Most of them have bars on the left and right side and a bonding jujmper that runs the width of the panel between them.

You can't just go willy nilly screwing bars to your panel. You need to use the ones appropriately listed for your panel.

No, I misspoke once. There is a neutral bar in there, which I have to check for bonding to the case. I need to add a separate ground busbar, not bonded to the neutral bar that is in place (the neutral is stood off by plastic, but could be bonded by screw).

Straightforward, the neutral conductors will have a busbar, and the ground conductors will have their own, where the neutral and ground busbar are not electrically continuous.

Umm... in your first post

So I'm guessing that I need to buy a neutral bar and screw it to the box so it's bonded to the chassis and separate from the neutrals?

and then in your second post:

The neutral busbar has black plastic standoffs, so if I drilled in a neutral bar I think I'll be set.

In any event, you appear to understand the issue, but note you can't just go drilling holes in and mounting bars. You need to use the appropriate listed means for the panel.

Yes, I understand the situation, sorry for the incorrect terms. Brain farts...

I'm still interested in what should the ramifications of this been, since fortunately I saw none.

The bars I've seen imply they have self tapping screws, so two holes do the trick. But I will be verifying. Too bad this old cooper box doesn't say anything useful on it's internal label.

Thanks again.

... if there are not separate ground and neutral wires in the cable feeding this subpanel, and it is very old, then you're not dealing with anything horribly catastrophic - that's just the way it was done, once upon a time.

Once upon a time is in excess of 30 years ago.