Feasibility to track and accurately time runners?

I’m interested in using these modules to track runners on a track to get accurate times. The use case would be athletes would wear the tag and an anchor would be placed at the finish line. Could it possible to accurately time athletes as they cross the finish line?

Yes but depending on the accuracy you want it may be overkill.

A class1 gen2 900 MHz RFID tag has a read range of 10 meters and is designed to allow for easy detection of things as they pass through a doorway. As such it would be good for detecting people as they pass the line. This has the added benefit of costing pennies per runner and not requiring a battery on them.

The down side is that you wouldn’t get the exact moment they crossed the line, only when the reader first saw them. Smart placement of the reader antenna should make these two very similar but it wouldn’t be perfect.

UWB would allow you to get more accurate but at a far higher cost and depending on how many people are crossing the line at about the same time still won’t be perfect.

Eventually I would like to be able to track the runners at all time to get various times at any point on the track. I imagine this would be possible with anchors set up around the track.
Would I be able to use the DWM1001 boards to implement TWR or will I need PDoA?

You could use two anchors on either side of the track.
It seems reasonable to assume the runner is traveling in a straight line at a constant height perpendicular to your anchors. With those restrictions you could use either TWR or TDoA to detect when they pass the line. That would allow you to make multiple measurements and interpolate to give you a fairly accurate time that they crossed the line.
You could also use a single PDoA anchor, simpler but a little less accurate. And a single point of failure. With two anchors if one doesn’t see the tag (e.g. due to other runners blocking it) then you can at least get an approximate time based on the second anchor alone.

You could do it with a single DW1000 anchor and just go by time of closest approach but that would limit accuracy to the measurement update rate. At that point there is no benefit over the RFID method.