Can I grant multiple remote LocalService accounts access to a SQL Server database?
Machine accounts such as LocalService cannot be trusted over the network.

There is essentially no way to authenticate the remote machine without domain authentication of some kind. 

Use SQL Server authentication; that's what it is meant for.

Alternately, you could provide a setup procedure that allows your clients to provide a domain account for each service to access the SQL Server database.  You'd need to provide a mechanism where they can enter domain credentials for each service when they are setting up the server end.