It seems like there are Haneda Airport Express Bus (limousine)operated by Keihin Kyuko Bus running between Haneda airport and Fujisan Sta.(Kawaguchiko).
Please see the link below for your information.
Unfortunately, the reservation page is only in Japanese.
If it is difficult to make an online reservation, you can also buy the ticket at the limousine bus counter in Haneda airport. However, April 30 is the beginning of Japanese long holiday, so all the seats may be already reserved. The bus leaves at 12:25 and 14:40.
Fujikyuko Bus is also operating a bus between Haneda airport and Fujisan station.
Fujikyuko Bus also operates a bus between Mt. Fuji area (Kawaguchiko) to Nagano.
Please book your seat from the following website.
You can make a reservation from 1 month ahead of your departure date.