You can load more than one internet browser on your Mac - I have Safari, Firefox, and IE on my MacBook, and have needed each one at some point. However, I haven't shopped for CORI since I got the Mac, so I don't know if my newer version of IE would work.
Since you're new to CORI I would only do one shop for them - take the easiest and least out-of-pocket one you can find - and see if you can send your report. If it doesn't, try another browser, and another. On another thread, some people say Chrome works for them.
If it works, I would next try one shop that requires photos, and see if your photos go through. If they don't at least you know it's not the browser.
If you have issues and the report won't send, at least you're only out one shop. Then switch to MSCs that don't operate in the dark ages.
NOTE: I'm not on the forum every day. If someone comments on my post, I might not reply right away. I've been a shopper since 1991. I've never done any work for a MS company in any other capacity.