Weapon level is locked by either Research Level or player level. In the case of weapons that can be researched in the Research terminal, you can only build up to the highest level you have unlocked.
For weapons that are unlocked by default, you can build the next tier up every time you reach a level threshold, which are the same thresholds that unlock the next tier of research upgrades.
For example, when I started, I could only build an Avenger 1. When I reached level 6, I was then able to build an Avenger 2. However, if I wanted to build a Widow 2, I'd have to research level 1 and 2 of the Widow and be level 6 or above.