The Authorized Statistics on Women's Age-Specific Fertility Rate (ASFR) in Japan, 1920-

The annual figure of age-specific fertility rates (ASFR), related indexes (TFR, NRR, etc.), and the basic statistics (female population of each age, number of live births) have been carried every year on the official journal of National Institute of Population and Social Security Research:
人口問題研究
= Jinko Mondai Kenkyu
= Zinkou Mondai Kenkyuu
= Journal of Population Problem
ISSN:03872793
NCID AN00120247

The figures are in the article with the title of "全国人口の再生産に関する主要指標" (Population Reproduction Rates for All Japan). The titles were slightly different before 1963.

Notes

The number of birth included only children with Japanese nationality. Figures were derived from Vital Statistics (人口動態統計), with special modifications to calculate age-specific fertility.
The age-specific female population is as of October 1 in each year, based on Population Census (国勢調査) (for years divisible by 5) or Population Estimates (人口推計). Some special modifications were made for the calculation of age-specific fertility.
For the figures before 1970, the age-specific female population included all women living in Japan, regardless of nationality. They are not comparable to those after 1971, which excludes non-Japanese nationals. Recalculated indexes (TFR etc.) were published in the 1972 report.
Data before 1939 are different those published in the official current statistics. I have no information about the cause of the difference.
The 1966 and 1967 data are with 5-year age classification. Other years' data are with 1-year age classification.
The 1975 and 1980 data are prompt reports using 1% extraction from Population Census. The final data are on the appendix tables in the next year reports.
Okinawa is excluded from 1947-1972 data. There has been no correction to keep comparability.
There was no correction made for leap years. Use the weight of 365/366, if you are conducting a high-precision comparison (viewing 0.005 or smaller difference) among years.

Recommendations

As for 1971 and later, you can think of the tables in the reports below as the officially accepted ones, except for the 1975/1980 reports (instead, use the appendix tables in the 1976/1981 reports)
As for 1970 and earlier, the tables in the reports had expired. You can use them only as approximate values (they are biased to lower fertility than today's statistics). I don't know there are open data for these old statistices made comparable to the current ones, but the original figures are available from the past reports of Vital Statistics, Population Census, and Population Estimates.

List and links

Before 1954: No. 64 (1956): http://www.ipss.go.jp/syoushika/bunken/data/pdf/14207005.pdf (上田正夫 + 山口喜一)
Before 1960: No. 87 (1963): http://www.ipss.go.jp/syoushika/bunken/data/pdf/14209303.pdf (山口喜一)
For 1961, 1962: No. 94 (1965): http://www.ipss.go.jp/syoushika/bunken/data/pdf/14210006.pdf (山口喜一)

For 1963: Not found
For 1964: Not found

For 1965: No. 104 (1967): http://www.ipss.go.jp/syoushika/bunken/data/pdf/14211007.pdf (山口喜一)
For 1966: No. 108 (1968): http://www.ipss.go.jp/syoushika/bunken/data/pdf/14211406.pdf (山口喜一 + 金子武治) * by 5-year age classification
For 1967: No. 112 (1969): http://www.ipss.go.jp/syoushika/bunken/data/pdf/14211806.pdf (山口喜一 + 金子武治) * by 5-year age classification
For 1968: No. 116 (1970): http://www.ipss.go.jp/syoushika/bunken/data/pdf/14212206.pdf (山口喜一)
For 1969: No. 119 (1971): http://www.ipss.go.jp/syoushika/bunken/data/pdf/14212508.pdf (山口喜一)
For 1970: No. 126 (1973): http://www.ipss.go.jp/syoushika/bunken/data/pdf/14213206.pdf (野原誠)

Note: Change in the data source for the age-specific female population. Re-calculated indexes (TFR etc. for 1925-1972) are included in the report for 1972 data (in No. 131 (1974))