This is one of the most common questions we get from South African clients, and the good news is: generally, no - as long as your website is properly hosted rather than run from a computer in your own office.
Reputable hosting providers keep their servers in professional data centres, which have multiple layers of backup power (UPS battery systems and diesel generators) specifically designed to keep servers running through load shedding stages and power outages. Your website, email and other hosted services should stay online even while your own office or home has no power.
What load shedding does affect is your own ability to work on your website, check email, or take calls about it while your power (and possibly your fibre/router) is down at your location. This is a good reason to have a mobile data backup for your own internet access, even though the website itself stays up.
If you ever notice your specific website going down in a pattern that matches load shedding schedules, that's worth reporting to your host immediately - it usually points to a specific server issue rather than being normal.