It is very possible, even without a wheel. The game has a considerable learning curve but you can always begin with the easier cars and keep some assists on. I play a lot of racing games, but I found Dirt Rally very challenging at first: I crashed a lot, and I couldn´t beat even the slowest AI.
110 hours later I mastered most of the non-RWD cars, got every achievement in the game, and put several times into the global Top-100 (played on Xbox One).
Absolutely rewarding game. Try it.
Yes yes yes. Do not start with a RWD car. Don't do it.
Otherwise play hyper conservatively. If you don't crash you'll do alright in the first championships. That's not as hard as it sounds, just, as you run your first races, never, ever go flat out. If you're playing with a controller feather the throttle and never depress it fully (I think this comes more naturally with a wheel). I'm only talking for a few races, but I really think learning throttle control/response is the biggest key to success in this game.
If you start getting through the tracks without crashing, you get a feel for the car (also, stick to one car to start). Pretty soon you learn when you can push it and when you can't. This was how I got good at RWD cars. Just going super slow and going slightly more aggressive as I felt more comfortable. I have a wheel, but play equally with both since sometimes I don't want to setup the wheel. I'm honestly probably better with the controller.
Dirt Rally is probably the most rewarding game I've ever played, and I've sank over 150 hours into it. It would've been more, but I started with a RWD car in early access and bailed on the game for 3 months before finding a proper starting vehicle. Don't do that