Alexandre's answer gives a good out-of-universe explanation: my answer is the in-universe explanation.
Alien3 takes place on the prison planet Fiorina 161. This is a prison-work colony for violent offenders. From Wikipedia:
The pod crash-lands on Fiorina "Fury" 161, a foundry facility and
penal colony inhabited by male inmates with histories of physical and
In the real world circa 2016, prisoners are barred from having weapons. I see no reason why it would make sense to give weapons to violent offenders currently serving a prison sentence even in the fictional future of this movie.
Why not provide weapons locked up so only the wardens have access?
This is an isolated, remote planet. It takes time to travel there: it appears that at the beginning of the movie a call would be sent out with the identity of the life boat, and it took the length of the movie (several days) for a response team to show up.
Looking at the prisoners, there are some big, tough dudes. Would you want to be the warden who can unlock the weapon room with those guys wandering around, when it would take several days for help to arrive? That is a recipe for mutiny, despite the themes of repentance and atonement in the movie. Having lethal weapons accessible to known violent criminals is too much of a temptation.
Besides, why do they need weapons to begin with? It is not every day that a bloodthirsty alien is on the loose, and a planet like that one does not appear to have much in the way of indigenous life that would be a threat to the colony.