Can people criticize the Imperial Family in Japan? I can't find a case or article about law, people in prison, or deportation of a foreigner for criticizing the Imperial Family.
The 1947 constitution abolished Lèse majesté in Japan.
Yes. Japan has adopted Free Speech and other western liberal democratic institutions following the 1945 surrender to the United States and subsequent occupation, though it wasn't the first attempt at some westernization. Beginning in 1868 with the Meiji Restoration and lasting until the 1945 surrender, Japan underwent a major period of industrialization and westernization, borrowing many concepts from Europe (Japanese culture is really a hodgepodge of "what they thought worked best". The legislative body is a Diet (German) but the overall government form runs like the U.K. (Constitutional Monarchy) while the Judiciary is a Civil Law system (continental Europe) with some nascent jury trial (civilian judges are sourced from the population and sit with career officials in making a determination at trial. This is much more recent development and is still being experimented with). Japan has a long history of adopting what they like from other nations, and the American Occupation brought some new innovations to the Nation, including American-style Free Speech (and more culturally, the complete abandonment of arranged marriages for dating that was common in the United States, though the dating-marriage time is notably shorter in Japan).
Prior to the Meiji Restoration and the Imperial Japanese period (1868-1945), the Emperor and his family had notably less political power than the title's meaning holds in the West or even Japan. The Imperial Family is thought to have descended from the Shinto Sun goddess from the creation myth and thus the Imperial Family is both culturally and spiritually important, but not politically important. In fact, the Meiji Restoration was the largest significant period of Japanese history where the Emperor had something to do politically. Prior to that, the military commander, the Shogun, was the person who actually ran Japanese government (with the Emperor's blessing), and the current situation vests most of the highest political powers with the Prime Minister (in fact, the Emperor of Japan has less power in his government than the Queen of England has in hers).
Additionally, pre-Meiji restoration, individuals did not have rights and the lowest organization with any legal rights in Japan was the family unit. Persisting to this day, Japan has a strong cultural taboo about making the in-group look bad to the out-group, be that group a family to non-family, office politics outside of the office, or even the nation to foreigners. Because of this, if there was any political gripe at all, it would have been discussed only within the household.
And since the family line is said to be descended from a god, insults against the Imperial Family would have been quite taboo, as Shinto's core beliefs are that spirits exist within every object and must be afforded respect, so it would be quite taboo to disrespect a god incarnate. In fact, the U.S. wanted to depose the Emperor as part of surrender, but they were informed about the status being sacred to the Japanese people, who wouldn't accept the Emperor abdicating, let alone forcibly, and they settled for his remaining in power with the provision that he renounce claims to divinity (but was still permitted to pray to the goddess as an ancestor on holidays where such acts are customary). Because of this situation, and the fact that the nation was under a military imposed government once again (this time it was foreign to the island as a whole, not an internal feudal war between warlords) there are some historians who jokingly (and some historians who seriously) refer to the occupation leader as "Shogun MacArthur" and consider him the last Emperor (and not a bad one at that, either). Suffice to say, it seems that if there was a law prior to 1945 about badmouthing the Emperor, it wasn't often violated, as the Emperor was respected by the people he ruled, and more often then not in history, the Emperor wasn't a political force causing misfortunes ... it was the local feudal lord.
This taboo also has one bizarre law with respect to Flag Burning in Japan. It is legal to burn the Japanese national flag in Japan, but it's not always legal to burn the flag of a foreign nation. Specifically, if the nation who's flag was burnt presses charges, Japan will arrest the individual responsible (assuming the burnt flag was legally owned and thus burnable by the responsible person). My understanding is the sentence is usually a fine. The United States, conversely, will diplomatically tell the complaining nation where they can stick the complaint. Again, this oddity can be ascribed to Japan's taboo about making the group look bad in front of outsiders.
Another oddity of Japan is that Yakuza are frequently associated with vans with political speakers (commonly used by Japanese Far Right Wing political activists) as the loud political propagandizing is perfectly legal and makes it hard for Police to quietly pull them over and not face accusations that it was for protect speech and not an actual crime. Real protestors typically will send apology notes to local residents a few days later, for disturbing them by their protests (I had family stationed in Japan but living off base (the section of town was very well known for having lots of Americans) and they would occasionally get a middle of the night wake up call from one such speaker truck, followed by a letter sent to them the following day, apologizing for the trouble, according to those who could actually read the Japanese characters. This was in Tokyo, and apparently, the situation was much worse in Okinawa, but they would also receive apology notes later that day.).