Crank axle cracked on 1972 Centurion, looking for guidance on how to replace
The crack is in the bottom bracket(BB) axle. This will at least need to be replaced. If the crank arm(silver part) is not damaged this can be reused. 

There are many types of bottom bracket standards but it is most likely this frame has a 68mm English thread square taper bottom bracket which are the most common in pre year 2000 bikes.

Something like a Shimano UN26. Any bike shop should be able to supply and fit a new bottom(price in my local area would be $30-50 supply+fit). You do need specific tools to remove and fit bottom brackets so unless you plan on investing in some tools I suggest get a shop to replace it.