Not around Shinjuku, but go to Ueno, and you got a veritable camp of blue tarp tents.  Also, in cold weather, the tunnel connecting Ueno JR Station to the Ueno subway station is regularly lined with miserable homeless old men.  It is shocking.  Also, around Shibuya station, there are two or three regulars you can count on seeing, most conspicuously around last train time: an old lady and an old man, as I recall.

More info on homeless in Tokyo

Having said that, the number of homeless in Japan is estimated by the Ministry of Health, Labour and Welfare to have declined from about 125,000 in February-March 2003 to 18,500 in January 2007 (Wikipedia).

Taking 20,000 homeless in Japan over a 127 million population, that is a 0.016% rate.

For comparison, as of 2005, 744,000 homeless in the U.S. over a population of 300 million, that is 0.25% -- or over 15 times as many homeless per capita as in Japan.

As of 2007, France had 100,000 homeless over a population of 60 million, which comes out to 0.16%: far less than in the U.S. but still ten times as bad as Japan's.

... all progress depends on the unreasonable mensch.
(apologies to G.B. Shaw)