Actually quite a few places allow you to get in without such deposits. I've lived here since my last two years of college - totaling 8 years now - and never had an issue getting in with almost zero down aside from rent and guarantor company fee. My rents have ranged from $350/month to now $1600/month.

On the private side: コンフォリア

On the public support side (subsidized housing): UR (Urban Renewal) JKK Tokyo (東京都住宅供給公社)