Tatoos are sign of Yakuza/Gangster membership
"This is certainly the most bizarre rule I have ever seen, and from my knowledge of Japanese culture, there is nothing wrong with tattoos. Anyone have any ideas? "
Traditionally, Tatoos are sign of Yakuza/Gangster membership. So, who wants gangsters in the place? This has chnaged a little bit recently in Japan with some young people getting tatoos, but they are are not generally socially acceptable.