You need access to the database and for that you need the owner. It happens a lot, the owner goes awol for whatever reason. The domain name was renewed in July but that might be a standing order thing they didn't notice, so the address at least will keep going for another year but the hosting might run out although you have no way of knowing that. You're better off cutting the cord and breaking away while you still can and you have the option to let people know. If the hosting deal ends and the site vanishes you'll all be none the wiser, won't know where the new site is and the posts will be lost anyway. While the board is working, now is the time to start a topic on there and get everyone involved in what the plan is. If you do want to save old posts you can have teams of people split in to forums and dates/subjects to go through copy pasting it in to text files.
As I said, realistically, how much of that older content do people really look at regularly?
If there was a script you could remotely run to steal the entire content of a board, people wouldn't need to struggle to get sites going, they could nab 100,000 posts from established sites to increase their page content and rankings without doing any work.
can't guarantee it'll work though as you might need to be able to read the forum content and as you can't change the permissions you won't be able to do that.