You want to continue using the built-in contact form functionality, but you also need to add a file field to get uploads from your customers. This is a very common issue Shopify Merchants face, and this app is how you can fix it.
Unlike other upload apps, this one:
After a very quick install, and for only $5/mo, you can add a file input to your form and links to the downloads will be embedded in the same email you're used to getting.
This plugin should work with any image (JPG, GIF, PNG) and any PDF.