MYSQL Trigger not working
The first problem is that you don't have an END IF;.  

The next problem (assuming that you aren't doing this and just didn't include it in the question) is that when you have a compound statement, you have to change the statement delimiter from ; to something else so that the client will send the entire CREATE TRIGGER statement to the server as a single statement.

Finally, the column "first_name" probably needs to be qualified with NEW (assume you're referencing a column in wp_usermeta called "first_name".

DELIMITER $$
DROP TRIGGER IF EXISTS update_customers $$
CREATE TRIGGER update_customers AFTER INSERT ON wp_usermeta
FOR EACH ROW 
BEGIN
  IF NEW.wp_metakey = NEW.first_name THEN 
    INSERT INTO test_customer(first_name) values (NEW.wp_meta_value);
  END IF;
END $$
DELIMITER ;