Once you take it apart, you’re unlikely to be able to buy new wheels. Better to replace the whole unit - I did this just last week with an office chair.
Pull the entire black part out of the gray part but don’t throw it away yet. Bring the old castors to the hardware store to check the size and shape when you buy new castors. In Australia, new ones are sold like this:
I would expect that the wheel shaft has a flange or maybe some knurling on it to prevent it from coming loose. You will probably need to drill it out. I wish you the best of luck on your journey, you will learn a lot about how things are made. I also expect that this will end with you buying a new castor or a new suitcase.
go to a big box store. go to the metal bar stock section- usually they're in the pick-and-choose nuts and bolts area. Alternatively, just ask. (I like local hardware stores, they''ll have it too, somewhere. better selection of pick-and-choose hardware, too.)
Get some smooth rod slightly smaller in diameter than the rod there. use the rod to knock it out- either in a press (if you have access to one) or by setting it in some kind of clamp and pounding on the rod with a hammer.
if their only options for bar stock are expensive, you can get a bolt that's also slightly undersized and use that.
for the record, the wheel looks like it's an absolute gonner, so you're probably going to have to get more parts anyhow.
I would suggest a bench vise. The beefier the better (but be careful not to clamp down too hard on the plastic.) One way to do it would be to use your typical woodworking clamps c-clamps, or whatever to press it against the top of a vice or a pair of blocks split enough to let the bolt/pin to fall through. the big thing is you don't want it flying anywhere when you start applying force.
a hydraluic press would push it out with more control on such a set up, but most people generally don’t have them. that's why I suggested the hammer-and-rod set up. what that will do is let you apply force diretly to the pin; and with it slightly undersized, it won't replace the pin.
Generally the entire caster wheel assembly would be replaced instead of specific parts of the assembly. these kinds of wheels are not made to be taken apart to be repaired- which is why you’re having all this difficulty. amazon has plenty of kits- but try and match brand to brand.
I assume the metal axis is probably molten firmly into the plastics with some kind of knurling. So probably you are going to break the plastics if you are trying to push it through.
As suggested by others, a solution may be to try to drill it out. The other option I see, is to remove the whole black part and replace it by new rollers which have standardized shafts are when dedicated for furniture.