Organ transplants were developed only in the last few decades, so they aren't mentioned explicitly in the Bible. But I find nothing there that would forbid them - and in fact there's much in the Bible to support them.
For example, the Bible reminds us that our lives come from God, and because of that, human life is sacred and we should do everything we can to preserve it. Do you remember the story Jesus told about the Good Samaritan? (You can read it in Luke 10:25-37.) It tells about a man traveling along a lonely road who was beaten, robbed and left for dead. Others ignored him, but one man (a member of the despised Samaritan race) stopped, bandaged his wounds, then took him to an inn and paid for his care. He did all he could to save a stranger's life - and so should we.
Talk over your desire to become an organ donor with your family; they need to be aware of your intentions. Then take whatever steps are necessary to register as a donor in your state. Recently the life of a dear friend of mine was spared through a heart transplant, which came from someone who had registered only weeks before as an organ donor.
The Bible does, however, talk about another type of "organ transplant" - and that is the new life God gives us when we come to Christ. The Bible says, "I will give you a new heart and put a new spirit in you" (Ezekiel 36:26). Have you experienced this in your life? You can, by asking Christ to come into your heart today.