NEW DELHI—A few years ago, Ravi Kumar made a big bet on the future: He sold his horse-drawn taxi and started driving an electric rickshaw instead.

“The times were changing,” he says.

Today, he’s going old school and switching back.

Here, in the chaotic lanes and bazaars of Delhi’s old city, customers aren’t ready to say nay to horse-drawn taxis quite yet. Known as tongas, a handful of these wooden-wheeled, one-horsepower anachronisms still ply the streets in several parts of town.

So earlier this year, Mr. Kumar, 36 years old, paid a visit to a third-generation carriage maker in Delhi who still makes a living—if only barely, he grumbles—building tongas to order.