I can't currently get a hold of a lot of sources on this, so I will only focus on one - rather glaring - problem that the Wehrmacht would have had in accomplishing such strikes. The lack of succifiently long range weapons or weapon platforms. There are only two ways to reach the US when coming from Europe, by air or sea. Let's take a look at the possiblities by air first:

The closest the Luftwaffe could possibly get to the US from a land base would probably be somewhere in occupied France. The distance between New York City and Brest in Brittany, France is approx. 5300km (or 3300 miles)

Missiles: While the Germans were a leading force in the development of missile technology (the V2 rocket is the most famous example of that), they had only a few hundred miles (lacking an actual source for this, but I'm pretty sure you can look up the correct values in T. D. Dungan: V-2. A Combat History of the First Ballistic Missile) of range. Not even remotely close to be capable of crossing the Atlantic.

Long-range bombers: One of the bombers with the longest range that I could think of the Focke-Wulf Fw 191. And this plane never really got out of the prototype stage during the war. However, it had a rather impressive ferry range of 3800km (Green, William: Aircraft of the Third Reich. Volume one). Still, not enough to reach the US (and even if they'd have managed to extend the range with additional fuel tanks, that would only amount to a one-way flight.).

The Luftwaffe was accutely aware of this problem and had plans for a long-range bomber that could strike the US (creatively named "Amerikabomber"). This project however never managed to get off ground - if you will forgive the poor aeronautical pun - during the war. In fact, Hermann Göring (as head of the Luftwaffe) only saw those plans for the first time in 1942 (for more on this check out Duffy, James P: Target America. Hitler's Plan to Attack the United States). At that point Germany frankly had more pressing concerns, such as the war against Russia.

This brings us to attacks by sea: The German Navy had proven on several occasions that they could strike quite close to the coastline of the United States, mostly through U-boats which hunted the supply convois and other ships. One of the most devastating was the so-called "Operation Paukenschlag" (Operation Drumbeat - also known 'American-shooting season'), which resulted in massive losses to the Allies. More than 600 (!) ships were lost, this translated to roughly a quarter of the total sunken tonnage by subs during the entire war. (Gannon, Michael: Operation Drumbeat.) This, however, does not translate into the ability to strike at naval bases (nevermind the land bases or factories where the tanks etc might have been produced). During the last days of the war, two German subs actually managed to get in the direct vicinity of the coastline. One was sunk a few kilometers off the coast of Rhode Island, the other surrendered a few days later at Fort Miles, in Delaware.

As far as I know there never were any attempts at serious strikes at American naval bases via surface vessels. I don't have any sources for this, but I imagine the sheer number of US Navy vessels and planes would have rendered any such attempts futile.)

Maybe someone else can shed more light on other factors, which prevented the Third Reich from striking at the US Mainland. For example, I would imagine that even gathering the intelligence on where to strike (in case of design bureaus for example) would have been tricky.