Depends on exactly when the entry occurs. Previous the British had been conducting discussions with the Portuguse government concerning Brit air & naval bases in the Azores, and plans were either Spain of Germany to invade Portugal. Let s imagine for a moment several scenarios
1. Spain is paid by German to invade Portugal to seize control of the Wolfram or tungsten ore. Backstory: Britain had been outbiding Germany for the ore. This was horriblly expensive for Britain, but it was impossible for Germany. The Portuguese & Spanish demanded cash or equivalent from Germany which was in very short supply. & it kept the Portuguse happy & friendly to Britain & the US.
2. Portugal & the US fear a Axis invasion & the Azores are occupied in a agreement similar to that allowing the US occupation of Iceland in 1941.
3. A coup by a pro Axis faction removes Salazars government. Fearing Axis air and naval units based in the Azores the US makes a premeptive invasion.
Any of these three could result in Hitler deciding to declare war on the US. He had toyed with the idea several times in 1941, but never followed through. Any of these three might tip the scale.
The earlier any occur the less well prepared the US is for such a operation. When the US mobilization started in latter 1940 the only ready amphibious force for the Atlantic fleet was a combined arms brigade of Marines and the ability to form a second such group. The Army had already started working to develop a capability for landing operations, but 'Amphibious Forces Atlantic Fleet' was a paper capability through much of 1941. By midyear the equivalent of two Marine brigades and a nominal division HQ existed, and a Army division had learned how to get on & off the boats and pack its equipment in waterproofing. Other miscl Army and Marine units were training for amphib ops, but were not well along yet.
Of course the Portuguese Army had only a few battalions on the Azores, so its not a challenging fight.