(Photo: Love’s Old Sweet Song by linda yvonne)
The syntax for doing this is similar to doing an update in one table based on values in another table yet simpler.
INSERT INTO suppliers (name) SELECT customers.name FROM customers WHERE customers.id = suppliers.id);
If you want to add constant values into the insert you can do something like this.
INSERT INTO suppliers (name, city) SELECT customers.name, 'Toronto' FROM customers WHERE customers.id = suppliers.id);
For further reading please see SQL INSERT INTO.