One way is to create a SYNONYM for Customer.dbo.Table1 in the Supplier database. You can then query the synonym as though it were in the Supplier database.
Suppose you have the Customer database with Table1 and the Supplier database with Table2. Suppose both tables are in the dbo schema. You can then test the following in SQL Server Management Studio (SSMS):
USE Supplier; -- Tell SSMS to switch to the Supplier database
CREATE SYNONYM Table1 FOR Customer.dbo.Table1; -- this creates a synonym permanently in the Supplier database; you only need to execute this once
SELECT * FROM Table1; -- select all from Customer.dbo.Table1, from within the context of database Supplier
This way there is no need for TRIGGERs and the corresponding overhead (you would need 3 TRIGGERs to keep the tables in sync).