Can a US or other foreign exporter take care of the Customs clearance in Canada?

Yes. This is very common and they are referred to as Non-Resident Importers (NRI).  This means the exporter, who has no physical presence in Canada, becomes the Importer of Record for goods cleared into Canada.   They then take responsibility with Canada Revenue Agency for all entry issues.    This includes all fees, declarations and penalties.   Because of the complexity involved in the rules and regulations of importing goods into Canada, having a customs broker to assist you is a smart business choice.