Although what you're describing is possible, it sounds like it might be overkill for the example you're considering.
Perhaps instead you could try
- Download a zip file of all the files in your email (often email providers offer this option)
- Use LabVIEW (or do it yourself?) to unzip the files into some directory
- Use a For loop and a list of files, generated by LabVIEW, to process the files one by one and store the results in whatever way you currently do this?
Directly accessing your emails introduces a whole raft of concerns with security (passwords etc), filtering (what if one of the emails you receive has nothing attached, or isn't related to the processing step you're describing?), network connectivity (not likely a problem, but could be...), and so on.