I'm not sure if I'm about to describe this correctly but I'll go for it anyway.

One of the tuner buttons on my Lag acoustic has worn away on the inside so that it no longer grips onto the metal part. The original buttons are too nice to change so I'm just going to change this one button.
First, is there any way of finding the original tuner buttons and buying them from Lag or wherever? On the internet I only seem to be able to find pretty naff looking ones.

Or instead, if I bought any tuner button of the same style (keystone) would it fit the original machine head, or would I have to get a completely new machine head?

you'd have to get individualized specs for your tuner keys as well as whatever tuner key you are looking into. but you should be able to find one that works for you.