Well you might be out of luck then, your best hope is maybe some japanese streaming sites with optional subtittles if any exist, or ask some jap to rip from disc. But you may as well watch it in finnish and english subs, it should be easier to find but apparently they too have been targeted recently https://www.youtube.com/playlist?list=PLhyvnlUyHTDtBj4OH8IGL2W8uA-Z5GfFG
Anyways, try searching it as "Tanoshî Mûmin Ikka" for better search results, and you can use sites like https://subtitlebank.net/moomin-english-sub-726788/
to try and peace together the subs.
Also theres this; https://www.reddit.com/r/LearnFinnish/comments/9o52xv/where_can_i_download_or_watch_the_moomins_cartoon/>That's for Finnish audio with English subs
I didn't look any deeper into that since it was implied to be finnish and we are searching for japanese, but there may be something useful there.
This is the closes that I found; https://www3.kissanime.pro/watch/tanoshii-muumin-ikka-bouken-nikki.7ymy/7586q7
It may be what you were looking for.