SQL - Trigger to update a column as NULL when another column has a fixed value
I agree with Andriy that a constraint should be used, but to answer your question if this is possible with a trigger, the answer is yes.

Here is the trigger:

CREATE TRIGGER i_Seller
ON Seller
AFTER INSERT
AS
IF EXISTS (SELECT NULL FROM inserted i WHERE Type = 'Org')
BEGIN;
    UPDATE s
    SET MidName = NULL
    FROM Seller s
    JOIN inserted i ON s.SellerID = i.SellerID
    WHERE s.Type = 'Org';
END;