Does my customer get an email when I edit their order?

No. Saving changes on the Edit screen updates the order quietly. No email or message goes to the customer.

The customer will see the updated items and total when they look at the order in My Orders on your store.

If the change affects the price or delivery, let them know yourself, for example by email or phone. Then add a note on the order (Add note) so your team knows the customer was told. The edit itself is logged in the Timeline as "Order edited by" with your email.