You should already have two records of a charge: the receipt email from checkout, and invoices inside the customer account. You do not need to open a dispute to get a copy.
Print invoices from the account
If you can sign in at https://app.backgroundchecks.com:
Go to Account & Billing → Billing & Invoices.
Print one purchase with the PDF button, or export everything with Export Invoices (Excel/CSV) on the upper right.
Full steps and screenshot: How do I print invoices?.
If you cannot sign in, use Forgot your password? on the login page, then come back here. See Do I need an account to order a report?.
Receipt from checkout
When you pay, BackgroundChecks.com sends a receipt email (subject you will see: Your receipt from BackgroundChecks.com). Search that inbox (and spam) for the date of the charge. That email is the receipt. Replying to it is not the fastest way to get a PDF invoice — use Billing & Invoices, or write support as below.
Ask support for a copy (not a dispute)
Email [email protected] or call 1-800-942-1366. Subject line something like Copy of receipt / invoice. Include the email on the account, approximate charge date and amount, and last four of the card if you have it. This is a document request, not a report-accuracy dispute, and not a chargeback.
Charge you do not recognize
Email [email protected] first. Do not treat an unrecognized charge as a consumer-report dispute.
Wrong information on a report
That is not a receipt request. Accuracy disputes go to [email protected]. See How can a dispute be opened? and Dispute the Accuracy of Your Background Report. Only the applicant can open a report dispute.
Unused order / refund
If you want the charge reversed because the applicant never finished, that is the unused-order policy (120 days), not a receipt copy. See 9437975.