You may need a new threshold plate.

Are you talking about the trim on the jamb moving, the jamb itself moving, or the threshold? Those are potentially different problems with different solutions.

1/2 inch is a pretty good amount of movement, if it is at the very bottom surface soul plate obviously something has either broken off or wasn't sercure. You will have to either pull the threshold and check the lower door framing or toe nail it pilot holes , then counter sink the screws. But I would start with pulling up the threshold

From what I can tell, it’s the jamb itself.

How is the jamb attached to the bottom part typically? I’m assuming this was supposed to happen but the construction is poor so wondering what the fix is. Is the what “counter sink the screws” means?

The jamb should be affixed to the framing and none of that should be moving. Get some long, exterior screws -- maybe 3" long -- drill pilot holes at a 45 degree angle near the base of the jamb, and use an impact driver to sink the screws into the floor surface.

I finally got around to removing the threshold. Turns out the jamb isn’t attached to the concrete at all. And what kind of subfloor is this? Thanks for your help!

Seeing that, here is what I suggest.

1. Get a tube of construction adhesive and squeeze it into that hole/gap I see at the base of the jamb, making sure the base is positioned exactly where it should be.
2. After the adhesive cures, you want to secure the fixed piece to the left of the jamb at the base. Drill downward and put a hole in at about 1" in from each end. You will need a hammer drill to go into the concrete. Use Tapcon screws with the correct size masonry bit and follow the directions. You can countersink the head of the screw into the wood piece and then finish it off with wood button plugs.

As for your subfloor, all I see is a strip of unfinished wood that appears to be part of the flooring. If you are on a concrete slab, there could be no subflooring.