Given the unanswered questions about the exact cause of the illnesses, many health experts have said that people should not vape. According to the C.D.C., young people and pregnant women should never use e-cigarettes, and adults who do not smoke should not start using them.

Still, a recent survey of thousands of students found that about one in four high school seniors had vaped during the previous 30 days; in 2017, one in nine seniors said they had done so. The researchers who conducted the survey concluded that new safeguards were needed to protect teenagers, as “the developing brain is particularly susceptible to permanent changes from nicotine use and when almost all nicotine addiction is established.”

In one of the suits filed on Monday, the Francis Howell School District in St. Charles, which has about 18,000 students, accused Juul of “taking a page from big tobacco’s playbook” to develop “a product and marketing strategy that sought to portray its e-cigarette products as trendsetting, stylish and used by the type of people teenagers look up to.”

Three Village Central School District in New York State also filed a lawsuit on Monday. Similar suits were expected in the coming weeks, said Jonathan P. Kieffer, a partner with Wagstaff & Cartmell, the Kansas City, Mo., law firm that filed the first suits .

Among the steps that schools have had to take against vaping, Mr. Kieffer said, were installing sensors in bathrooms, removing bathroom doors, banning flash drives, hiring more staff and paying for programs that help students deal with nicotine addiction. The suits did not specify the amount of damages the districts were seeking.

“In numerous meetings that we have had with educators, administrators and board members across the country,” he said, “the consistent theme that we have heard is that districts do not intend to stand by and allow this crisis to escalate but instead are taking a stand to protect their kids.”

Legal experts said that though the suits were reminiscent of litigation against makers of guns, opioids and tobacco, the districts could face a stiff challenge in establishing their case.