If you are the sole proprietor of a business you are wearing many hats and what's simple when life is 'normal' can be nigh impossible in a storm.
As a small business owner, I would ask someone to email customers on my behalf. And be honest about delays if they are happening. Being sick isn't the issue; being less than honest about shipping dates is. You don't stop being responsible for people's money because life happens. At least not if you want to stay in business afterwards. (I have been unexpectedly sick. I have called customers from the hospital to apologize and cancel work that I was expecting to be able to do. That's life when you are the sole proprietor of a business.)