Getting patients to the doctor's office is a big cost for many insurers. That presents an attractive opportunity for Lyft and Uber, which to date have focused on consumers.

Non-emergency medical transportation is a $6 billion market, with most of that money going to cover the poor and elderly, who often don't have cars or can't drive. Medicare and Medicaid providers typically foot the bill.

CareMore Health System, which offers plans to those populations, started working with Lyft last year so caregivers could schedule and monitor rides for their patients.

Now the insurer is spending less money on rides and saving time. Within a three-month period in 2016, CareMore said it reduced wait times by 30 percent (from 12.5 minutes to 8.8 minutes) and the cost per ride decreased by one-third from traditional services, including taxis.

"We've saved more than $1 million in one year by shifting to Lyft," said Sachin Jain, president and CEO of CareMore, in an interview with CNBC.

To appeal to people without smartphones, Lyft built a web-based tool for caregivers, nurses and patients to schedule rides and track them in real-time. Rides can be scheduled more than 24 hours in advance.