How can ionic substances have an infinitely-repeating lattice structure?
To be technically accurate, it's "practically" infinite.

It's not infinite since as you mention a grain of salt (or even a large crystal) has some finite size.

You have to remember the actual atomic dimensions $10^{-10} \:\mathrm{m}$. So while it's not truly infinite, there are an incredibly large number of atoms in a crystal.