How to real time replicate a SQL Server DB without primary keys defined
It is unlikely that any replication solution you can import would work without a PK (or at least a unique index) otherwise it can't tell the difference between an update and a delete+insert (amongst other problems) which could be quite a significant issue depending on what keys and triggers exist on or refer to the affected table(s).

If you can't add a PK or other unique index then you will need to try write your own synchronisation solution for the situation.

Is there any reason why the table(s) can't have a primary key?