I have never been asked to fill out a captcha when logging a case. Possibly its a new feature they are implementing and haven't got it quite right yet. I have been able to update an open case of mine without any issues
Are you able to call support for the time being, may also pay to let them know.
This might be a UI bug. I'd recommend just giving support a call (help>get help) for your best support number and asking support.
I've had no issues submitting cases.
As luck would have it, I ran into this exact issue this morning submitting a case.
I received the same error about the CAPTCHA. In my case though, the 'I'm not a robot' box appeared below the case description field and I was able to check the box.
It appears that Meraki is using Google's reCAPTCHA in an invisible mode. I'm guessing that if you don't 'pass' the check when you submit the case, the checkbox is supposed to appear, so that you can assert your non-robot-ness.