Does my customer get an email receipt for an online order?

Only if you switch it on. Stripe makes a receipt for every successful payment and refund, but emails it automatically only when you turn that on in the Dashboard.

• Go to Settings, Business, Customer emails and turn on Successful payments and Refunds.
• Nothing is sent when a payment fails or is declined.
• Stripe needs the customer's email address, and the refund email also needs the customer stored on the charge.
• You can still send a receipt by hand from the payment in the Dashboard.