Is it possible to replace a broken drive axle on the 4-8-4?

Assuming you mean an axle with driving wheels.  Maybe you mean a drive "shaft".

Yes, if:

1.  You have the tools to remove the side rods, valve gear , etc.  2.  You are able to remove the broken axles from the drivers without breaking  anything else. 3.  You are able to securely mount the new axle onto the drivers properly "quartered ".  And get the  worm gear on too,.  If you can even get a replacement axle, unlikely.  4.  You are able to get all the running gear rods, etc, Back together again, which even with long experience is not a walk in the park.

Or.  You can get a replacement driver set; again, if you can find one, unlikely.

It's way more work than anyone can imagine to successfully home repair Model steam locomotives.

Plus, the modern model railroad world does not have many spare parts.

Usually,  a new engine is the most cost effective route.

I suppose it is if you are mechanically capable. I did not see any parts in the parts list though.
You could check Shapeways as they sell a kit for Bachmann 4-8-4 loco axle repairs.
Possibly you could return the loco with a check for a replacement. I have done that but contact Bachmann first. Look at the Information page.

https://www.shapeways.com/product/PJUUQ9266/bachmann-ho-4-8-4-axle-centers

Rich