Damicci,
The first problem in the sentence is that you've translated "drift toy" cars directly into Japanese.
Drift cars are certainly shortened into ドリ車 , but it doesn't mean "drift toy" is ドリ玩具 .
玩具 is "toy" as a manufactured product viewed from the business/industrial standpoint. So, the word is not appropriate.
Since I know nothing about this sport, I've checked for a Japanese equivalent for the vehicle but could not find anything in Japanese about "drift toy" cars. I think you should check this out with someone.
Also, "vehicle" is 乗り物 but "norimono" points to anything a person can ride, any mode of transport including trains and airplanes. When you mean anything in the "car (on wheels)" category, you use 車（くるま・しゃ）
Coming to the structure of the sentence, it is not a simple question, since the problem lies in determining the best order of modifying phrases. I think you are getting yourself into complex sentence structure:
"A person riding in a "drift toy" is dragged around a building by means of a cord attached to an off road vehicle."
建物の周りに｜コードで｜オフロード乗り物に｜取り付けた｜ドリ玩具に｜乗ってる人を｜引きずっ た。
It can be translated into something like this:
人を乗せた（ミニチュアの）ドリ車が、オフロード車に綱（ケーブル）で引っ張られて、ビルの周りを回った。
The subject has been changed from "person" to "car" because it sounds more logical to me.
Please note that this isn't perfect, and I'd very much prefer to say it in an entirely different way to sound natural...
This is all the time I have for now!!