No, I find nothing in the Bible that would prevent you from being an organ donor, and if your family agrees with your decision, then I would encourage you to do so.
In fact, doing this could be an important way to help someone facing a serious medical condition. Although you would never meet that person in this life, think of what your donation could mean to them. At least it could give that person a better quality of life–and in some instances could even mean the difference between life and death. The Bible says, "Therefore, as we have opportunity, let us do good to all people" (Galatians 6:10).
Your question, however, touches on another issue that concerns me greatly–and that is the number of people who are killed or seriously injured in traffic accidents. One reason you were asked to indicate on your driver's license if you were willing to be an organ donor is because tens of thousands of people die on our roads every year–and you could be one of them. Ask God, therefore, to make you a better driver. This is just as much a God-given responsibility as any other.
Above all, remember that organ transplants may delay death for a time–but Christ offers us "an inheritance that can never perish, spoil or fade–kept in heaven for you" (1 Peter 1:4). Is your faith and hope in Him?