That would depend on the type of thyroid problem.
Prospective blood donors with a history of current or previous thyroid disease due to an autoimmune disease process (that is, due to the formation of antibodies directed against one’s own body tissue) would be advised not to donate blood.
In general, donors who have an autoimmune condition (such as ankylosing spondylitis, autoimmune thyroid disorders, rheumatoid arthritis, inflammatory bowel disease, etc) are advised not to donate blood.
Our restriction on donors with autoimmune disease is mainly to prevent any complications in our blood recipients. Autoimmune disease can cause some immune system disturbance/dysfunction. There is a small risk of precipitating similar immune system disturbance and symptomatic disease in those receiving blood from donors with autoimmune diseases.
One of the commonest causes of thyroid disease in young adult females would be autoimmune thyroid disease (such as Grave's disease).
Similarly those with a history of thyroid cancer would also not be eligible for blood donation as a precaution.
Donors with a history of non-autoimmune thyroid disease (such as a thyroid nodule) should be able to donate blood if their thyroid hormone levels are normal and they are not currently on anti-thyroid medication.