Yes, thyroid hormone replacement hormones require a prescription. They aren't supplements or drugs, they're hormones.
You will have blood work prior to attaining your first prescription, and if your doctor is any good, will most likely have you do blood work every 6 weeks (or there about) until your hormone levels are right for you and stable, then you'll be dropped to every 3 months on the blood work. If things remain stable, you'll probably be dropped to every 6 months, at some point for blood work. Some doctors only require blood work annually, if you're stable and feel well.
My doctor writes my scripts for 90 days, with 3 refills; that's 1 year. I doubt I'd get by without having blood work at the end of that year, except for maybe a month or so extension. Any good doctor is going to require regular blood work...