I would say arthrogryposis is usually diagnosed by a doctor noticing multiple stiff joints and limited movement, often at birth, and sometimes even before birth on ultrasound. After that, the main steps are a physical exam, medical history, and X rays. Depending on the case, doctors may also order genetic testing, bloodwork, nerve or muscle testing, or sometimes a muscle biopsy to figure out the cause.
The main specialists I would want involved are an orthopedic doctor, a physical therapist, an occupational therapist, and often a geneticist. A neurologist or rehab doctor can also be very important depending on the person.
The big thing to know is that diagnosis is not just about saying someone has arthrogryposis. It is also about figuring out what is causing it, because that helps guide treatment and long term care.