We don't scan anything. For us it's much easier to pull up the PDFs of the EOBs from our insurance website (or the Rx claim from our pharmacy plan). Each month I save the EOBs to a folder on our computer, and send myself an email to my own Gmail, and put it in a specific folder on Gmail for healthcare receipts. We then submit receipts as needed, always submitting the oldest first. That way I have two copies saved electronically, one on our Mac and one in the 'cloud'.
As discussed on another thread, it may be best to pay cash for medical expenses IF you are able to max out ALL your tax advantaged spaces, and let the full contribution balance grow in the HSA. If not, it may be best to submit your receipts, THEN chunk as much as u can in your IRA or 401k to further reduce your taxable income. Anything left over in the HSA can be left to grow as well obviously.