Can I export several invoices from one customer's account at once?

Currently, there is no option to mass export invoices for specific customers. You do have a couple of alternative options, however:

1. If you have Account Login enabled for Hosted Pages, then you can simply provide the Account Management URL to the customer and they'll be able to login to their account at https://yoursubdomain.recurly.com/login and view all past invoices on their account. You can find their unique account management URL in their account page under Hosted Pages.
2. You can download the Invoices - Summary export and then use the account_code filter to only view invoices for a specific account code. Keep in mind that this will not download PDF versions of the invoices, but rather an Excel file with the contents listed here. 
3. You can also fetch an Account's invoices via API. For Version 3 API, here and, for Version 2 API, here.