How can I get a thyroid blood test at home?

https://www.medicare.gov/care-compare/results?searchType=HomeHealth&page=1&city=Seattle&state=WA&zipcode=&sort=alpha

Here is a link for Home Health agencies in your area. Chose one and call them. They will not come for blood work only. If you have another reason that is covered they will also do blood work.
Perhaps you are dizzy and need your BP monitored and perhaps some PT to make you stronger. Perhaps you could use help setting up your pill planner?
As Barb said, call one and let them come for an evaluation. They know what is covered by Medicare. Sometimes an elder will say they don’t need help when asked and that is the thing that can get them approved. They will contact the doctor for the proper paper work. make it easy for them to help you.
I am assuming here that you have Medicare insurance.

Or you can call your local area agency on aging and ask for help with transportation to a lab or other services.

https://www.agingkingcounty.org/

You can read about it more here:

https://www.medicare.gov/coverage/home-health-services

If your doctor certifies you as homebound, you are then entitled to the Medicare Home Health service. You need to be recertified every 60 days.

It sounds like your doctor doesn't know this. Consider calling a large, reputable Home Health Care Agency (like the Visiting Nurse Service) and ask how you can access this service.