Hi Howard, your cabinet is, as you say, Japanese and dating from the mid Meiji period when such things gained enormous popularity with both the gaijin in Japan and worldwide via a buoyant export market. Some can be mind boggling quality.
Yes I have (now retired) spent years restoring such things (as a hobby, not a business) for some of the UK dealers. Missing inlay is obligatory on these! I used to actually carve whatever was necessary….be it shell or coral or ivory etc but it is very time consuming and therefore rather expensive. You need a specialist workshop set-up. I doubt you will find anyone to actually carve replacements nowadays but another trick is to search for simpler panels (made in their 1000s) and very often heavily damaged and thus very cheap. It is sometimes possible to “harvest” elements of inlay that can be adapted far more easily to either fit in or even stuck straight over the top of missing areas. Flowers, leaves etc are common and thus quite straightforward. Faces are a real pain, they hardly ever fit, usually looking the wrong way or the wrong size which usually means a lot of hard work or a vaguely acceptable compromise.
The best advice I can offer is to look for panels (auctions etc) in the hope of “do it yourself” repairs.