Old H&R 999's had what was called the "Rice grip frame", Rice being the name of the man who designed it. It is curved at the back so that a wide variety of grip shapes can be attached to it with one screw in the back edge of the grip. The gun in the photo is that style. It has been out of production since 1950's at least, so find good wooden grips to fit it will be tough. I would do a Google search for gun auction websites and look on them; I'd also check E-bay.
The grips it has on it are a good quality early plastic. I think it is called Tenite or cellulose acetate. Do not keep the gun in anything very airtight; if it is cellulose acetate, it gives off tiny amounts of acetic acid, not enough to hurt the gun, but enough to cause the grips to disintegrate. Do another Google search for "vinegar syndrome" and you'll see what I mean.