You can buy "organic fertilizers", as PP say such as compost, manure, etc.
However I say that with a grain of 'salt' because fertilizer is organic to begin with. Fertilizer is a balance of salts and minerals.
You need to start with your garden bed and build a solid veggie garden. You can get dirt delivered of you do not have great soil.
Then you add the organic matter (peat moss or stuff in bags, like manure, compost, etc) and "fertilizers", like bonemeal (calcium), bloodmeal (slow release nitrogen), magnesium (epsom salts), gypsum, etc.
Research the veggies you want to grow and see their requirements.
If you want to grow veggies you have to "feed them" and spray for bugs. There are "organic" bug sprays you can use out there as well as other ways of dealing with insects.
Key is always to research what you are growing. Look up your local extension agency. They usually have great tips and things like disease and bug warnings.
