Yes, a productive vegetable garden can be grown with just water and nutrients (called hydroponic gardening) but that is a specialized way of growing without soil. Some garden soils are rich and free draining and will produce well without compost and mulch. Mulch  is used to keep down weeds and seal in moisture so hand weeding and watering can be done instead. If your soil has a lot of clay or sand then compost is the best way to improve the tilth or texture. Compost and mulch are not fertilizers. High yield vegetable gardens need nutrients as over the years plants take out the natural nutrients and they must be restored. Compost can supply some nutrients but it depends on what the composted material was made from. Fertilizers restore nutrients to the soil especially the big three of nitrogen, phosphorus and potassium. Keep growing, Marianne Binetti