Note: Most races can be applied for through Runnet. You can also contact the race organizers and get apply-by-mail application forms. You usually need to have an address in Japan to apply. Most races cannot be entered on race day. (Although some, run by running clubs, can.)
For more information on entering races in Japan, head on over to the Japan Running News blog at http://japanrunningnews.blogspot.jp .
Their specific page on entering races in Japan is here: Race Entries.
Now for a short list of races that you may be able to apply for in Hokkaido.
Nikkan Sports Toyohira-gawa Half Marathon (Sapporo)
May 5 (Children's Day)Dist: Half, 10k, 5k, 3k
Application: Feb 1, through March 31
http://www.toyomara.com/
Hokkaido Marathon - 30th Anniversary (Sapporo)
August 28Application starts April 3 at 18:00pm and closes shortly thereafter (usually, within two hours!)
If you want to try and get into this race, and it is worth it, visit it's Runnet site at
https://entry.runnet.jp/entry/runtes/user/pc/competitionDetailAction.do?raceId=123074&div=1
Or their homepage for more information. Also, if you are registering online you need to create an account with Runnet. You will need a credit card and an address in Japan. The marathon also has a multi-language website with information.
http://hokkaido-marathon.com/2016/overview-en/
This is a new marathon started in 2015 and is one I am eager to try.
Application starts April 1 and runs through June 30.
http://www.abashiri-marathon.jp
Ohotsuku - Abashiri Marathon
September 25This is a new marathon started in 2015 and is one I am eager to try.
Application starts April 1 and runs through June 30.
http://www.abashiri-marathon.jp
Northern Horse Park Half marathon and Ekiden (Chitose / Sapporo area)
May 14 and 15
Dist: Half, 7km Trail + Half, 2.5km
Application until March 31
Karikachi Trail Running (Shintoku / Tokachi area)
May 22
Apply by April 25
20km, 12km, 4km Trail Run
Apply on Runnet: http://runnet.jp/entry/runtes/user/pc/competitionDetailAction.do?raceId=120793&div=1
There are many more events coming up, especially in June. Check the archives to get an idea of where and when races are!