Can I rent a motorbike in Nikko?

I'd like to drive the Irohazaka route

You can in Utsunomiya although you may well need to either be a Japanese speaker or have one with you. https://www.rental819.com/english/area/area.php?area=2

Do you have a driver's license that you can use in Japan and a credit card to rent a motorcycle?
If yes, you can but if not, you can't.
https://www.rental819.com/english/first-of-all/
http://nankai-hotline.com/rentalbike/rental_toi.html
Credit card will not be used unless it is an international brand.

I do not know if you are a foreign tourist or already living in Japan.
If living in Japan, you should check if this can be done or not.
https://youtu.be/IRDayfpjAbQ
https://www.police.pref.kanagawa.jp/eng/e_mes/eng83000.htm

I guess they know well.
http://www.gaijinriders.com/forum.php