It's not simple....
The simple version is as Rod says. An appropriately approved LAA inspector can sign-off an engine rebuild. Also as Rod says, you can also do it yourself and have an inspector sign-off at critical stages. I have done several myself on this basis.
The more complex version is (by my understanding) that if the engine is an 'approved' type (e.g. Lycoming, Continental etc.) then you can only use 'approved / official' parts to rebuild it. Quite how you prove that the parts are approved / official without Form 1 (or US documents) is not really defined, but custom and practice seems to indicate that little or no paperwork trail is required. If they look like original parts and your inspector is happy to sign them off, away you go. My understanding (and I'm happy to be told otherwise) is that you can't just get on your mill or lathe and make new parts yourself.
Modifications to parts (e.g crank grinding) needs to be done according to official specifications. There is however no requirement to use an approved facility.
The better LAA inspectors (I can point you to one or two via PM) work to very high standards and will not sign off anything that would not be suitable for use in a CoA engine.
In the case of non approved engines (VW, Subaru etc.) then the situation is a bit more confused. There is no source of approved parts so you have to satisfy your inspector that what you are using is of an acceptable standard. Whilst your suitably qualified LAA inspector can sign-off for a rebuild, the LAA sometimes take quite an interest in what you are up to. I think it stems from a time when poorly converted engines (VW in particular) were not as reliable as one might have liked.
In any event, you will have to complete a comprehensive build log, detailing all the bearing clearances, ring-gaps, parts used etc.
I'd suggest a quick call to LAA Engineering.