Need for Speed Unbound

Time effective solution

The cheapest car available at the start of online the career mode is the Volkswagen Golf GTI (1976) for $27,000. You can buy multiple of these cars for your garage and each will count towards the achievement.

Cost effective solution

If you have finished 30 or more playlists you will unlock the ability to purchase the Volkswagen Bettle (1963) for $18,000. These can also be bought multiple times.

Once you have popped the achievement, you can sell the cars for money to put towards others cars, performances upgrades, or garage upgrades for the “Access All Areas” achievement.

This achievement requires you to own 10 vehicles in your online garage.

There's nothing wrong with the other solution posted here, however it would save you a lot of time and effort to do this after you've 100% completed story mode. That's because all the free vehicles you earn in story mode also carry over to online. This applies to the cars you earn while completing special races, qualifiers, the grand, and the activity and challenge reward cars. In fact, you'd earn enough free vehicles to unlock this without having to purchase a single vehicle online if you're going for the 100% completion. This does not, however, apply to vehicles you purchase in story mode. That means you can't bring your sweet Bugatti or Koenigsegg with you unfortunately, but no big deal. Below are the vehicles you can get for free that are tied to achievements:

Qualifier/The Grand Cars:
-2015 Land Rover Defender 110 Double Cab Pickup
-2008 Ford Crown Victoria
-2017 Nissan GT-R Premium Edition
-Your starter car from the prologue of story mode (mine was the 1988 Lamborghini Countach 25th Anniversary)

Activity/Challenge Reward Cars:
-1981 BMW M1
-2016 BMW M4 GTS
-2015 Porche Cayman GT4
-2018 McLaren 600LT
-2015 Nissan 370Z Nismo
-2016 Ford Focus RS
-2015 Ferrari 488 GTB
-2017 Lamborghini Aventador S Roadster

There are also vehicles you can win throughout story mode from specific races, however I believe these are optional and potentially missable. Here are the ones I won for reference (I believe this is all but one of them):
-2009 Lamborghini Murcielago LP 670-4 SV
-2006 Lotus Exige S
-1990 Ford Mustang Foxbody
-1970 Plymouth Barracuda
-2016 Alfa Romeo Giulia Quadrofoglio
-2007 Mitsubishi Lancer Evolution IX
-2018 McLaren 570S Spider
-2019 Dodge Charger SRT Hellcat
-1964 Aston Martin DB4

You can also earn A$AP Rocky's custom 1988 Mercedes-Benz 190E 2.5-16 from completing his mini quest line within story mode.

So as you can see, you can earn up to 22 cars for free (not including the one you first purchase online), and 12 of these are tied to achievements required to 100% the game. All I had to do was load up online and the achievement popped immediately without spending a penny online, plus you can use all of these vehicles in multiplayer races which is a huge bonus if you're looking to be competitive online.

If you choose to go the same route, you'll unlock your achievement as soon as you load into your online garage.