Thanks for the question. As you noticed the Session data included in the checkout.session.completed event does not include the line_items where the Price ID is associated to the Checkout Session.
line_items is one of the expandable properties, so to retrieve the Price ID you'd retrieve the Checkout Session and use expand to include the line items in the response. There is not a way to configure your webhook to have the data sent to you include this data.
There are two approaches to associating a customer's purchase with a Checkout Session. First, you could store the ID of the Checkout Session in your database alongside the cart or list of items purchased by the customer. That way you if a checkout session is successful, you can look up the cart by ID and know which items were purchased.
Alternatively you could listen for the checkout.session.completed webhook event, then when you receive a new notification for a successful checkout, retrieve the Session with expand then use the related price data.
Using stripe-node that would look like the following:
const session = await stripe.checkout.sessions.retrieve(
'cs_test_xxx', {
expand: ['line_items'],
},
);
// note there may be more than one line item, but this is how you access the price ID.
console.log(session.line_items.data[0].price.id);
// the product ID is accessible on the Price object.
console.log(session.line_items.data[0].price.product);
To take this a step further, if you wanted more than just the ID of the product, you could also expand that by passing line_items.data.price.product which would include the line items their related prices and the full product objects for those prices.