Which blood test do I need to get done to check the vitamin deficiency/hormones, thyroid in my blood. Do I need to doctors prescription to get it done?

TSH. Get a test done called TSH reflex which tests for thyroid stimulating hormone and only if abnormal they will automatically also get thyroxine or t3/t4 levels done. A relevant vitamin to check is vitamin d. Yes most labs will ask for a doctors rx and that way insurance will also cover your test.