Would the Japanese have been able to take over part of Mainland America in WWII?

It wouldn't have been feasible. It's telling that the Japanese high command didn't plan a follow-up to Pearl Harbor and as far as I'm aware never discussed the possibility of hitting it with a ground invasion. At the time the bulk of Japan's war effort is tied up in Manchuria and the Far East, which is not going nearly as well as anyone in Imperial General Headquarters would like. In all likelihood, they wouldn't have had the troops and certainly wouldn't have had the supplies to spare. Much less would they have been able to actually coordinate an effective line of supply across a hostile Pacific Ocean and then across a hostile continent.

I'm going to get a bit off topic here, but its important to understand the Japanese POV for the Pacific War. The United States had started an embargo against Japan in response to Japan's actions in Manchuria, which was severely impacting the Japanese economy and ability to conclude its war efforts there. Removal of the embargo was priority number one under Prime Minister Fumimaro Konoe's government in the mid to late 1930s, and there were several diplomatic overtures made to bring this about, which failed. It was after Konoe retired/was forced out that the thought process turned to war with the US, but there was a pervasive line of thought in the Japanese military circle that the embargo of today will become an American-led war against Japan tomorrow, therefore Japan must strike first before the carriers end up on our doorstep. The war with the US wasn't (as much) about territory as it was about removing America's ability to interfere with Japanese efforts in Manchuria and the Far East and thus complicate an already-delicate situation. Hence the focus on the Philippines and Pearl Harbor: remove the nearest Army garrison, remove the US Pacific Fleet and hopefully its carriers, and thus remove any ability for America to act offensively against Japan in the short-term.