Why wouldn't he have just run as a Democrat if he wanted to get into this race? He probably wouldn't have won, but there's no way in hell running a write-in campaign is going to work.
There was a Democratic Party establishment candidate, former Seward Mayor Edgar Blatchford but he lost the primary to perennial candidate Ray Metcalfe, a Republican State Representative from 30 years ago who in the interim founded and led the "Republican Moderate Party of Alaska". Even though Blatchford probably wasn't going to win, he would have ensured normal Democrats were represented in the race. Now that there is no longer that assurance, people want Begich to run.