Japanese funeral chanting in Tokyo Monogatari

The Buddhist temple in which the funeral scene takes place is Jodoji. It is a temple of Shingon Buddhism, and the chants that are shown during the funeral scene are thus a part of the traditions and practices of Shingon Buddhism.

The temple is identified as Jodoji in the essay "Buddhism in Tokyo Story" by Kathe Geist, published in the Cambridge Film Handbook "Ozu's Tokyo Story", edited by David Desser. The author discusses various aspects of how Buddhism is portrayed in the movie by Ozu and it is worth reading in its entirety. On the temple shown in the funeral scene, this is what the author writes in a note (on page 116):

11. Although there are few shots that might positively identify the temple in the funeral scene, Keiko McDonald believes the funeral temple is the temple seen from the harbor in the opening montage, that is, Jodoji, on the basis of the Japanese script, and the Tourist Department in the Onomichi City Hall has confirmed this identification. See McDonald's Cinema East: A Critical Study of Major Japanese Films (East Brunswick, NJ: Associated University Presses, 1983), 205.

Based on your comment under the question, it appears that your interest is not in Buddhism chants in general but particularly the one shown in the movie. Regarding this, I have found very little information. A YouTube search for Shingon Buddhism chants turns up very few genuine hits, and certainly no funeral chants.

In fact, this is what Wikipedia has to say on secrecy in Shingon Buddhism:

Today, there are very few books on Shingon in the West and until the 1940s, not a single book on Shingon had ever been published anywhere in the world, not even in Japan. Since this lineage was brought over to Japan from Tang China over 1100 years ago, its doctrines have always been closely guarded secrets, passed down orally through an initiatic chain and never written down. . . .

From the lack of written material, inaccessibility of its teachings to non-initiates, language barriers and the difficulty of finding qualified teachers outside Japan, Shingon is in all likelihood the most secretive and least understood school of Buddhism in the world.

This part of the Wikipedia article is not explicitly sourced, though. However, it does suggest that gathering more specific information is not easy. As I suggested in the comments, maybe you'll have better luck at Buddhism SE?