How much water and when to water vegetables depends on a number of variables.
The type of soil you have, ( i.e, clay, sand, silt, loam etc.), how much organic matter is in the soil, how windy it is, is there a mulch , when it last rained and how established the plants are.
As a guide seedlings will require more watering than well established plants, this is because the seedlings do not have a large root systems, another way of testing is to dig down about a spades depth and the soil should not be dry.
Given the correct soil conditions most vegetables will crop well with little or no additional watering
Vegetables and their water requirements
Requirements for watering vegetables varies depending on the crop, and growth stage. Leafy crops need more water than those grown for their fruits or roots. Seedlings will not recover if they dry out because of their small root system; water regularly in dry weather until established.
Vegetables - Drought Tolerant.
This is a list of vegetables that once established are pretty well drought tolerant and require very little if any additional watering.
Beetroot, Brussel Sprouts, Carrots, Chicory, Endive, Jerusalem artichoke, Leeks, Marrows, Onions, Parsnips, Radish, Spinach beet, Spring Cabbage, Sprouting Broccoli, Swede, Turnip, Winter Cabbage, and Winter Cauliflower.
The above list is not exhaustive bit gives a good idea of the type of vegetables that you will hardly need to water if the soil conditions are correct.
Vegetables - Regular watering (only in dry weather).
Leafy crops may need regular watering in dry weather, particularly on light soils to encourage leaf growth.
Vegetables - Water after flowering (only in dry weather, if needed).
Here is a list of vegetables that once established only need to be watered after they have flowered as watering before flowering will only encourage leaf growth and not flowers.
Beans (Runner beans need a lot more water than French beans)
Squashes, Pumpkins and marrows (many will survive without much extra watering unless you are trying to grow huge fruits)
The above advice only applies to those vegetables grown directly in the soil there are different requirements if you are growing your vegetables in containers.
Vegetables - Watering vegetables in containers.
Because of the limited amount of space and hence soil in a container the soil will dry out much faster than the ground. It is therefore necessary to water vegetables in containers significantly more often than otherwise needed. e.g. a tomato plant grown in a pot may need watering twice a day during hot weather. You maybe surprise to here that you may still need to water your containers even if it has rained, especially if there are plenty of leaves as the rain will bounce of the leaves and hence may miss the soil all together.
Vegetables - When to water.
You should water by applying the water directly to the soil either early in the morning or last thing at night, when the soil and air are cool, this helps to reduce evaporation and if any water has splashed onto the leaves helps to avoid burning. If you are growing crops that are susceptible to slug damage only water in the morning, this is because slugs are nocturnal and love moist conditions, so will be more active if you have just watered the soil.
For ways on how to reduce the need for watering please see Top Tips For Conserving Water in the Garden