Can customers download an invoice from their account on my store?

Not from their account. My Orders shows each order's items, totals, payment and tracking, but there's no invoice download button.

Customers get an invoice PDF attached to their order confirmation email, so ask them to check that email first. That PDF uses a standard layout.

If they need your branded invoice, send it yourself:

1. Open the order in Orders → Manage Orders.
2. Click Print invoice. Your active invoice template opens with the print window.
3. Choose Save as PDF as the printer and save the file.
4. Email it to the customer.

The printed invoice uses the template set active under Settings → Store Settings → Invoice Templates.