Ferry from Yakushima to Tanegashima?

I would like to visit both Yakushima and Tanegashima during summer of next year.

It seems that a rented car is the best means of transportation on both islands so I would like to receive my car at Yakushima airport and drop it at Tanegashima airport.

As far as I know there is a ferry from Miyanoura (Yakushima) to Shimama (Tanegashima). However, I do not find any web page for reserving a ticket. Even the company that operates the ferry has no home page.

Is a reservation unnecessary even if I go by ferry with a rental car from Yakushima to Tanegashima? Must I reserve by telephone? Are there any other ways but phone?

https://www.tykousoku.jp/

I can find a website for the ferry. But doesn't look like it can take cars.

http://www.furusato-tanegashima.net/access/taiyou.html
It is operated by Yakushima town. Check the latest info. But perhaps it is cheaper to rent a car in each island than ferry it to different islands including the drop off charge, if any.
We used the bus and the 3 day bus pass to visit all the places in Yakushima. The Yakudon restaurant and camp site (800yen/night) is just a few min. walk from the Anbo ferry terminal and a big supermarket and Anbo downtown cwnter/river..

Do two rentals - quite normal. Less common is to transfer cars between islands.