My understanding is that Jewish Law doesn’t allow organ donation. How is it that Judaism can place such an emphasis on health and pikuach nefesh and at the same time doesn’t allow organ donation?
Your understanding is incorrect. In broad terms- organ donation from a deceased donor to a patient waiting for a transplant (and not just to a "bank" to see if such a patient materializes) is usually permissible. Donation of a kidney from a live donor is usually permissible(but not required). Transplant of organs like the heart and liver- depends on the chances of success of the operation for the recipient and on whether the transplant would involve hastening the donor's death. In any given case a halachic authority MUST be consulted.