If you are just running motors it probably doesn't matter and a lot of other equipment can be jumpered t0 run 208 or 240.
Like John says, this is a PoCo option. They will need to add another transformer, probably replacing the small one you have too. The typical center tapped delta uses a smaller transformer for the wild leg and they only use 2, leaving one side of the delta open (AKA Delta Vee) Expect a fee for this.
Everything up to the panelbd will be replaced (the entire service). There is no question on the line side. I assume (I am going over to look at it tomorrow) that only the 3 phase loads are connected to the middle bus (in this case maybe a rooftop unit - it is retail). So when it becomes 208V the middle phase will be un-balanced. In addition - the panelbd is currently the service entrance. That will no longer be the case after the change (there will be a line side main disconnect). So the neutral needs to be separated (I'm thinking out-loud). Any other considerations?
You can swap some breakers around and balance the service once you get real 3p wye in there. You can get a head start by going in with your clamp when they are running their normal load and tagging the current on each circuit. Then you will have a starting point for the rebalance. We used to have to do that in computer rooms sometimes because each machine might have significant single phase loads and the luck of the draw might get some unwanted imbalance.