Well, the US military wouldn’t have even tried to send anyone at that time.
Likely the easiest way would be as a journalist associated with a major newspaper.
Prior to the US involvement in the war, US journalists actually accompanied some German armies.
As German troops entered Paris following their victory in June 1940, the American journalist William L. Shirer observed that they carried cameras and behaved as "naïve tourists." One of the first things Hitler did after his victory was to tour occupied Paris, where he was famously photographed in front of the Eiffel Tower.
But, by occupied, you mean after June, 1940. Note that a couple of thousand American expatriates who already lived in Paris remained there during the four years of occupation. A few thousand more had fled along with tens of thousands of French citizens, prior to Paris falling to the Germans.
So another possibility would be a relative of one of the ex-pats that remained in Paris coming to assist the person due to their becoming ill of something. They’d have needed to make contact with the German occupation to make arrangements, as they controlled the government and issued and tracked residency papers. You’ll want to look up books that discuss the German administration of Paris, the various offices and determine the appropriate office to ask for permission
But here was an official warning from the US government, issued in May, 1940, telling all Americans who hadn’t already fled to leave Europe.
The Germans used Paris as a mass tourist destination during the Occupation, shuttling in tens of thousands of their troops on leave over the years of occupation. Other than that, tourism wasn’t ‘officially’ blocked, and French citizens did travel within the country, although they needed passes to do so.
But even where international tourism wasn’t officially prohibited, the German “U-boat” war in the Atlantic made travel by sea exceptionally dangerous, as no ships were immune to being attacked. This more than anything that discouraged anyone without absolute necessity to cross the ocean to avoid doing so.