With an HSA, you just reimburse yourself by writing a check or doing an online transfer. The bank doesn't generally get receipts or care about the reason for the payment. You don't have to do the reimbursements expense by expense; you can collect and do a bulk transfer at the end of the year (or many years later, to let the HSA grow)

At the end of the year, you'll get a form 1099-SA from the HSA bank indicating the amount that was distributed from the account. You then declare that on form 8899, along with a line indicating the amount used for qualified medical expenses, which is hopefully the same amount to make $0 taxable.

You will need the receipts if the IRS audits that qualified medical expense amount. It's a good idea to scan them in and keep a tracking spreadsheet, especially if you delay reimbursement for years.