I'm not sure if these are equivalent to American W-4, W-2 and 1099, but after reading your post they are what came to my mind:
(1) 源泉徴収票 Gensen choushuu hyou (Tax Withholding Certificate)
It is a form to certify the amount paid (payroll, pension, retirement bonus, fee, etc.) by a business proprietor (incl. employer) to an individual (including employees) from which income tax was deducted.
It's mandatory for business proprietors to issue it and submit to the tax office as well as to the individual under Income Tax Act (article No. 226)
National Tax Agency Japan
国税庁
About Tax Withholding (pdf)
国税庁ホームページリニューアルのお知らせ|国税庁
Tax Withholding Certificate form (pdf)
国税庁ホームページリニューアルのお知らせ|国税庁
(2) 給与支払い報告書 kyuuyo siharai houkokusho (payroll report)
Employers have to submit the report on the payment made to each of their employee to the municipality (city, town, or village) where the individual is residing under the Local Tax Act.
Based on this report, resident tax is levied.
Payroll Report (pdf, on the left)
http://www.city.kasugai.lg.jp/dbps_data/_material_/localhost/12500/t1253000/200_Kojin/H22kyuhou.pdf