I scan physical receipts as PDFs, and store them in a Dropbox folder. I also have a separate cloud backup of my dropbox.
I always pay HSA out-of-pocket with a rewards credit card, so that I get cash back or points in addition to the credit card receipt. I also download the PDF of the credit card bill and store it with the receipt as proof of payment as well. For the few items that I pay by check, I download the check image from my bank and store it too. For stuff that gets billed online, I download a PDF of the bill from the medical provider (I have regular shipments of DME, for example, and save those bills). Same with stuff I buy off Amazon (first aid supplies, bandages, and condoms, for example, are reimbursable) or Zenni (eyeglasses). Ideally, for everything reimbursable, I have both a copy of the receipt and a copy of my payment in digital format in addition to the paper receipts.
I also scan as PDFs any prescriptions I get from my doctor. Because he will tell me to take an OTC medication--pill, vitamins, cold medicine, etc--and if he writes a prescription, it's reimbursable from the HSA in the future.
My Excel Budget Spreadsheet has a separate tab for reimbursable medical expenses. I note the date, the vendor, the amount, and the method of payment (which credit card/check/etc). I have a little note in the next two cells for when I save a digital copy of the bill and the payment.
At the top of the page is a running total of everything I've paid out so far which has not been reimbursed. I use that number as part of the calculation of my Emergency Fund, since now that I've paid for items out-of-pocket, if my water heater explodes or I need a new roof or if I'm out of work for a year and need to pay the mortgage, I can take money out of the HSA up to that total at any time, claim reimbursement, and use it like cash to help me out in a pinch (obviously only if I've gone through other accounts first... I think of my HSA as about level 4 or 5 of the Emergency Fund).