Absolutely.
In fact, it doesn’t work unless it is organic.  The entire system is connected, so trying to fix an issue for one part of your aquaponics system will affect the whole thing.  Adding chemicals, antibiotics, or other artificial additives to your aquaponic fish tank would eventually end up harming your plants. If you use pesticides or growth stimulants to aid your plants, it will harm your sensitive fish.  Every element that you add to your system must be plant-safe, fish-safe, and definitely nothing that’s anti-bacterial!
In terms of official USDA Organic Certification, commercial aquaponic farm are eligible to be certified.  There are strict standards that farm managers must follow, which regulate nutrient additions, pesticide use, food safety, and other aspects of farm operations.  For more information, reach out to your local organic inspector.