The ways and extent to which Jews have used vegetables in their cuisine have been influenced by where they lived (climate), their worldliness as merchants, and their dispersion and persecution. The Ashkenazic Jews of Eastern Europe largely knew only root vegetables (such as potatoes, onions, carrots), which could be stored through the cold winters, and such vegetables as cucumbers and cabbage, which could be pickled and stored.
Meanwhile, their Sephardic brethren enjoyed the colorful and plentiful vegetables of warmer climes, which they used in myriad and inventive ways. And, when some of their numbers underwent forced conversion, under the Inquisition, they traveled as merchants to the New World, bringing back a whole new range of vegetables which were quickly adopted into the Sephardic kitchen. These were adopted, in turn, by the others among whom they lived, especially as the Sephardim were dispersed through the Mediterranean basin, into the Balkans, and parts of Western Europe.
Separately, the Jews of Italy introduced certain vegetables (such as eggplant and artichokes) to new regions, as they were expelled from Sicily. Such dishes as caponata, so common to Italian cuisine, are often designated "alla giudia," in lingering recognition of their source in the Jewish kitchen.
In fact, many vegetables or salads which are prepared with oil, or a combination of vinegar and sweeteners, likely derive from Jewish cuisine. These were born of the need to prepare foods for the Sabbath in advance, so a form of preserving the foods was crucial, in the absence of refrigeration. Thus, we see the addition of sugar, raisins, other fruits, and spices such as cinnamon to vegetable dishes, to offset the vinegar used for preserving.
Along with this broad range of available vegetables and their use, others are eaten because of tradition and even puns. Leeks, for example, are associated with Passover, for the Children of Israel ate them in Egypt and bemoaned their absence during the wanderings in the desert. Carrots are strongly associated with Rosh Hashana, because the Yiddish word for them, "mehren," is punned to mean "plenty."
Vegetables are, thus, more than a sidedish in Jewish cuisine. They are a central component of it.