With the caveat that we are not kombucha experts, we think kombucha is OK for Passover.
Here’s why: Foods that are normally kosher year-round are also kosher for Passover unless they are made (or contaminated in commercial preparation) with wheat, oats, barley, spelt, or rye. If one or more of those five grains is used, then the food requires special preparation, e.g.. The concern of Passover is not fermentation (wine, cheese, and pickles are all kosher for Passover), but rather fermentation of these five specific grains.
That said, many Jews will eat only those commercially prepared foods that are kosher certified, and we have not yet found a kombucha product with kosher for Passover certification.