How to find the correct replacement glass dome for a light fixture?
See if the globe is a standard/common design.

There are a few "standard" designs for how globes/domes/shades attach to the lamp fixture.  Many manufacturers intentionally build to the standard, and there is a huge aftermarket of various styles of globe, specifically for replacement or "style upgrade" in any compatible fixture.  A lighting distributor or real hardware store will have these.  Big-box chain stores maybe not so much.  

Contact manufacturer

Part number is probably on a sticker on the fixture itself, usually around where the light bulbs are or on the back. You may have to take the fixture down to be able to see this (and if you do, be sure to turn the power off).

Just a warning: It's possible they don't sell the individual part, don't make it anymore, or want to charge more for the part and/or shipping than other options (below).

Buy an identical fixture

Find the same fixture somewhere, and use the parts from it you need

Just replace the fixture with something better

If you can't find a matching fixture, just replace it with something else.

You can also just use this as an excuse to replace it with something you like better anyway.