What type of doctor can order a swan ganz catheter placement?

Used much less. The swan-ganz (pulmonary artery) catheter is used to try getting data from the left side of the heart by measuring a "wedge" pressure from the pulmonary artery. Recent studies questioned its use, and technologies such as echocardiography, arterial-line instruments, and many less-invasive cardiac output tools are becoming more common. Eventually, it will not be ordered much at all.
Why? Cardiologists, pulmonologists, anesthesiologists, and cardiothoracic surgeons use swan-ganz catheters which are devices that measure pressure in the cardiac chambers and pulmonary capillary bed (which reflects left atrial pressure, the most accurate determination of heart failure.) but sg catheters are rarely needed except during heart surgery and are rarely used outside of the or nowadays.