I have a few squaddies armed with beam weapons currently because they do a lot more damage than conventional weapons. However, I don't seem to be getting nearly as many weapon fragments as I used to. Does killing an alien with a beam weapon vaporize the enemy weapon completely, preventing you from getting weapon fragments from them?
No. The only thing which will impact whether you get weapon fragments is killing enemies with explosives (grenades/rockets). The type of weapon you kill them with has absolutely no impact.
If you're getting less fragments over time than you're used to, it could be you're far enough in the campaign to be regularly fighting Chryssalids and Berserkers, which don't drop fragments when they die (since they don't have weapons). It could also be that you've just got enough to spend fragments on that they're getting scarce, which isn't generally the case in the early game.