One way that not irrigating helps a crop is by reducing weed pressure.
Ideas for growing without irrigation
Drought resistant plants (corn,wheat)
Use cacti to store watering an organic form. This material can be fermented in a closed vat for a few weeks to liquify it and make it's water soak into the soil.
Banana is another common option for solid water storage.
Pit or trench planting.(walipini,zai holes,waffle garden.
Decomposition or digestion. This process creates water from CH²O²∆ CO2 H²O. The trick is keeping the process going in cold or dry times and I guess the reaction has to happen near the plants root zone.