It is possible for a customer to attach files to a product before purchasing so the store administrator will be able to download those files when processing the order. It can be achieved with the special type of product options - 'Upload file'. In your control panel (Catalog → Products → Product management page → Options), you can add an 'Upload file' field to the products that require such attachments, so that a customer will be able to upload their files on the product details page before adding a product to the cart. You can also make that field required, this way your customers will only be able to put the product into the bag after they attach at least one file to that option. In the storefront, it looks like this:
The files that your customers upload via the 'Upload file' box are then stored inside Online store, attached to each corresponding order. You can get the exact files uploaded by the client for the corresponding order in the control panel → My Sales → Orders tab, on the details page of the particular single order you are processing. This screenshot illustrates it:
Please, note that the upper limit for a single file uploaded by a customer is 15MB.