Nebraska controls Ashiok, Dream Render and Alaska casts Goblin Engineer.

Can Alaska choose to shuffle their library, or is the whole process forbidden since they cannot search their library?

My apologies; I missed a more relevant Gatherer ruling:

If an effect says “You may search your library . . . If you do, shuffle your library” or “You may search your library . . . then shuffle your library,” your opponents can’t choose to search, so they won’t shuffle.

So, No, Alaska may not choose to search, so Alaska can't shuffle when Goblin Engineer's ability resolves.

d:^D