If you do find their site works on a other browser (and I mean, the browser, not due to an addon, etc.) then complain to the company / bank / etc.
If they take your complaint seriously then I'd persevere with them... if they don't then they're not customer focused and your business (money) goes elsewhere.
I think the main thing is for you to try doing a test restore of your data before you need to (and you already have a local backup anyway if your test goes wrong)
That will give you a better understanding of the whole process - they might be 100% reliable in storing data which is totally unusable by you because you've lost your decryption key, weren't backing it up correctly, etc (for example).