Yes, but obviously you'll need to make the logo yourself. Once you've done that, highlight all the layers (click left stick and A) you want to save, then create a new layer group. Name it whatever you want and then you can add that group onto any car you like. I recommend making a huge one and saving that, then scaling it down afterward (the grouped version will scale x and y equally). That way if you want a bigger one later you won't lose the quality. I save almost all of my parts that way, so that I can add them to any car later if I choose to.
And good news for fans is that they recently announced downloadable content in the next couple of weeks. New tracks and car packs are to be made available. No idea if it'll be free though, but I wouldn't count on it.