Well, the first reported car theft in history took place in June 1896 in France, when a guy named Baron de Zuylen had his Peugeot (I believe it was a Type 15 Peugeot) stolen by his mechanic. The vehicle was soon recovered by police a couple towns over and returned to the Baron.

One could also reasonable argue that the above is inaccurate, and that in fact the first car theft was actually committed by a woman named Bertha Benz (yes, she was the wife of Carl Benz, the same Benz from the famous Mercedes-Benz coupling). Back on August 5th, 1888, she reported stole her husband's new invention - a Benz Patent Motor Car Number III - from the family garage in Mannheim, Germany, Travelling at a hefty top speed of about 14.5 km per hour, she, along with her two children Eugene and Richard, drove the car (powered by a 2.5 horsepower engine) about 90 kms to Pforzheim. In that case though, she actually swiped her husband's ride because she wanted to take it on a marketing "test drive" to show the world that the vehicle was indeed reliable (something her husband was reticent to do at that stage of development). While her husband (Carl Benz) was initially pretty pissed off at her stunt, he soon warmed to it, as it indeed ended up showing the world that cars could be reliable forms of transportation over relatively long distances.

The biggest hinderance to early widespread threat of car theft was resale value of the stolen vehicle - whether it was selling the car "whole" or "parting it out" (stripping it and selling the parts individually). You need a customer for stolen cars/parts to make it a profitable enterprise worth the risk of getting caught of course, and market demand in the early automotive days was not very well established, meaning there were not a lot of buyers for stolen cars or car parts initially, because the use of cars was not very widespread. Parts for repair, and mechanics skilled in the repair, were hard to come by, so it took awhile for a criminal "black market" in stolen cars and car parts to pick up steam. It really was not until the advent of the famous Ford Model T - which first entered production in 1908 - that car theft, as OP asked, became "popular" (ie. "profitable for criminals"). The Model T was the first mass produced affordable car, with about 16.5 million of them being sold over the lifetime of production (which lasted until 1927). The parts on the vehicle were largely interchangeable throughout the lifetime of the vehicle's 19-year production run, so criminals discovered they could run a "chop shop", where they stripped a stolen Model T and could sell the parts out piecemeal to allow owners to replace damaged components and bodywork.

Keep in mind that by 1913/1914, 48 percent of all cars on the road in America were Model T Ford's (over 248,000 of the model were sold in 1913 alone), and lots of people who owned one got into accidents with their Model T's, as driving was a totally new experience for them, and took some getting used to. From minor "fender benders" to fatal wrecks, that meant that there were a lot of people who needed to repair their damaged cars, and that time period is really when the "black market" for stolen automotive cars/parts began to grow pretty dramatically. In just the city of Detroit alone for example, in 1917 there were 65,000 cars on the road, and that led to 7,171 reported accidents (plus no doubt many minor unreported collisions), with 168 of those accidents resulting in fatalities - meaning that over 11% of all cars on the road got into a reported accident of some form in Detroit in that year alone. That is a lot of cars in need of repair parts (whether parts were legally purchased or from the black market), especially when extrapolating those numbers across the entire US of A.

Also, The Model T did not even really require a key to start it until 1913, so early versions of the vehicle could be started by anyone via turning on the coil box and hand cranking it. Even once the basic ignition/coil box key was introduced in April of 1913 for the Model T, that feature was not too difficult for serious criminals to bypass (hotwire). Plus, back then, many car owners simply left that "ignition" key (which at the time was just a "skeleton key" that could be used to start any Model T) inserted in place in the car, as it was only really used to turn on the coil box to feed electricity to the engine, and was not literally used to start the car up, as there was no electric start feature at that time (I believe that the "electric start" feature for the Model T was introduced in 1919).

So, if you are looking for a period when car theft became somewhat "popular" (profitable for thieves), I would say it was in the early half of the 1910s when car theft became somewhat of a concern for those early automotive owners. It was not until 1919 that Ford introduced an actual ignition key for the Model T with true vehicular security in mind, though again, that was not too difficult to bypass, and thieves who swiped one of those very popular cars could quickly take it to a secure location, strip the vehicle of its parts, and sell the parts to unscrupulous mechanics who would use the parts to effect repairs on damaged cars.