how much access do Western (English-speaking, primarily) historians have to Japanese primary sources?

Are you asking how easy is it for foreign historians to gain physical access to Japanese primary sources or are you asking to what extent have Japanese primary sources been translated into foreign languages?

This is a really good question and there are a few things going on here. First of all, I'm not a historian and I don't read Japanese, so this answer will also explain how I go about things as an educator who stumbled on a lot of peoples hunger for reliable answers about Japanese history. I began explaining the academic work I was consuming, and here I am five years later. I'm sure a real Bakumatsu historian could do a better job at answering questions, but I like to think I know how to explain and teach.

i. The first issue is why so many questions are answered with reference to Western observations. This isn't because there are no Japanese historical accounts, or that English-language historians of Japan don't have access to primary sources. Its because Western observers had very different priorities for what they recorded than did Japanese people. Westerners like Louis Frois or Engelbert Kaempfer were interested in recording everything about the customs of Japan, their clothing, hair, material goods etc. and how they differed from the West. A lot of this is the stuff that their contemporary Japanese counterparts would have seen no reason to document. There is, of course, plenty of evidence in Japanese sources for these subjects, as well as the archaeological record and survival of art and material culture, but these Western ethnographies are a really good eyewitness report of things that often aren't addressed head on. So, for example, Louis Frois wrote

Because of our buttons and lacings, we cannot easily keep our hands close to our bodies; since Japanese men and women are not thus restricted, they always leave their sleeves hanging empty and pull their hands in close to their bodies, especially in winter.

This observation of a behaviour that was completely normal to anyone in Japan over most of its history isn't the sort of thing that'd make it into a Sengoku chronicle.

Western accounts are Japan seen through the eye of a stranger, and may be confused and biased and wrong, but they are really valuable just for bringing that perspective of a stranger.

ii. There absolutely is a lack of translated Japanese texts and translations of Japanese historical works. Even if translated, many texts are very diffiult to access. This isn't the biggest problem for historians. To be a real historian of Japanese history, I think you should be able to read Japanese. Professional English language historians communicate with their Japanese counterparts, work together on projects, and learn from each other. International cmmunication among historians has lots of problems, and often the English language and Japanese historians don't communicate as well as they could, but there's not a void between English and Japanese history. If you pick up any English academic book about Japanese history, it'll cite Japanese work, thank Japanese colleagues etc. Respected English language scholars do research in Japanese archives; they don't stay at home and just read translations of Japanese work. Luke S. Roberts, for instance, is a master of the Tosa domain archives.

But for the English-language reader, the gaps in the translations are huge. Important Japanese historical work, if it's ever translated at all, is translated many years after. There are so many untranslated primary documents. The amateur history-lover or undergraduate student is definitely restricted. This is my position. I'm in the very lucky position of having access to a university library and its databases, but I only read English. I'm not a historian, though I've had some good training from history professors and a talent for remembering and searching up stuff. Answering questions here to the best of my ability involves using English resources. This can be rather frustrating when I know from conversations or google-translate that there is an easy answer to be found in Japanese materials, but I don't have enough knowledge to properly reference it.

The translation gap is much more significant for older eras. There is a lot of stuff about the Edo Period in English, never as much as I'd like, but translations, analysis of data, amazing articles and books about many subjects. For the Sengoku, on the other hand, good up-to-date English academic work is way scarcer. Historical research in all areas of Japan is proceeding at a fast clip, and I'd say translations are coming out and becoming more widely available (via ebook format etc.) so the future looks bright, but it's still a challenge.

iii. There are of course way fewer primary sources as you go backwards in Japanese history, and the type of sources preserved will depend on the era's priorities. Even a Japanese scholar immersed in medieval sources will be very aware of the large gaps in the record. A closer example to our time: we have many, many records of the Late Edo Period, most of which take no interest in the ordinary lives of women. To reconstruct their lives, we have to search for sources, and tease out information from censuses, legal proceedings, satirical literature etc. Writing around the unknown, piecing together clues to construct fragments of an uncompletable puzzle, is the work of the historian.