Ah😅Understood your pain point.
The site has English translation by automation, then linked only Japanese reservation page(なっぷ) only.

To make reservation, you need to sign in なっぷ
If you already have the account of なっぷ, you can proceed as below. ( I guess you already have it, because you logged in this community)

1) choose the date from calendar
2) select the site type
3) tap the blue square at the bottom of right corner
4) select how many people use, planning check in time and fill in the blank of next page

But I recommend that you search other parks because the staff can’t speak English, it makes trouble easily….

Hope you can enjoy Japanese camping without any trouble.

This is the contact information for the reservation site, Nappu, so if you ask there you may be able to receive some kind of support.Enjoy Japan!
https://www.nap-camp.com/contact

舞洲バーベキューパーク⛺

U can make a reservation
by tel(06-6147-5858)maybe☺️