TEL AVIV—Amid Israel's tech boom, the government and many of the country's biggest tech employers are scrambling to bridge a gap between those who write JavaScript and those who read scripture.

Israel's tech-savvy defense industry and its small internal market have combined to help the country become a bigger player in the global tech industry. That has led to a bevy of startups and given rise to a community of tech entrepreneurs. Meanwhile, a wave of deal making, public listings and venture-capital funding has made Tel Aviv...