Real mother-of-pearl is hard to get any more. The types of shellfish cultivated and the average size at harvest, as well as the larger size of handgun grips preferred today, make it virtually impossible to get pieces of sufficient size to make grips out of, but there are scales available from custom knife-making supplies sources.
Mother-of-pearl is brittle and needs a backer in order to not chip. Many MOP grips from the early 20th Century did not have backing and would split or crack when the grip screws were tightened. Chipping is also common, as you have seen. You can epoxy a thin backing to the grips you have to make them less likely to chip or crack. Use sandpaper to work the material, avoid power tools as they put dust in the air, and anything that was once alive (including wood) can cause allergic reactions and may not be good for you.
MOP also loses its luster as it dries out, and polishing occasionally can restore shine and depth of luster to grips that have a dull look.