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The Apple II is one of a few computers that you could build a brand new one from parts if needed, they're not complex machines.

or run on an emulator

the article makes it clear that you need to interact with special cards to control the exhibits

That problem could probably be solved using an FPGA (or even simpler tech)

Given the specs, an AVR could do it.

How easy is it to find people with the know-how? How much do they charge?

I'd think you'd find dozens of qualified volunteers willing to work pro-bono to keep a Apple ][-powered museum running.

How about 10 years from now?

You'd find slightly fewer.

Still safe to leave the old stuff running if they get the job done and you don't want to add any new feature to it. And make sure you build your new stuff on a future proof platform.

It really depends on the roadmap.