Using Triggers in SQL Server
Two main ways, neither involving a trigger (generally it's better to avoid triggers if you can) -

1) Use an OUTPUT clause with the first INSERT statement to pick out the generated identity value:

DECLARE @NewOrder TABLE (order_id INT);

INSERT INTO Orders
    (customer_id, track_num, pckg_num)
OUTPUT
    inserted.order_id INTO @NewOrder
VALUES
    (@customer_id, @track_num, @pckg_num);

INSERT INTO Items
    (manuf_name, model_name, qty, notes, order_id)
SELECT
    @manuf_name, @model_name, @qty, @notes, order_id
FROM @NewOrder;


2) Rather than declaring Orders.order_id with the identity attribute, create and use a sequence object for it (requires SQL Server 2012 or later):

CREATE SEQUENCE OrderIDSeq AS INT
    START WITH 1;

CREATE TABLE Orders (
    order_id INT
        PRIMARY KEY
        DEFAULT (NEXT VALUE FOR OrderIDSeq),
    //... other columns as before
);

GO

CREATE OR ALTER PROCEDURE AddOrder(
    @customer_id INT, @track_num NVARCHAR(50), @pckg_num NUMERIC,
    @manuf_name NVARCHAR(50), @model_name NVARCHAR(50),
    @qty NUMERIC, @notes NVARCHAR(MAX)) AS
BEGIN
    DECLARE @order_id INT = NEXT VALUE FOR OrderIDSeq;

    INSERT INTO Orders
        (order_id, customer_id, track_num, pckg_num)
    VALUES
        (@order_id, @customer_id, @track_num, @pckg_num);

    INSERT INTO Items
        (order_id, manuf_name, model_name, qty, notes)
    VALUES
        (@order_id, @manuf_name, @model_name, @qty, @notes);
END;