I believe you need a lockpick of 100 to get the pulse gun.
No you don't. There's a key at Nellis. There's even a quest marker leading to it.
The triggers for the quest are seriously messed up. I tried half of them and non worked. I thought it was because I didn't proceed with the main quest (as is the case with Arcade) but then I got a trigger at someplace that had nothing to do with the quest. That's when I realized I hit a bug or something.
The easiest way to initiate the quest is (assuming you're on PC) take Veronica to Gibson's Shack, unlock the computer through console and use it.
But that's just part of the problem because there are possible glitches at:
1-Finding tech. Sometimes she won't realize you have the Pulse Gun and quest won't proceed.
2-Elder talk. She won't initiate conversation with McNamara and/or you.
3-Paladin's at the end. Veronica becomes stuck.
To whoever coded this quest: I've got a mini nuke with your name on it. One of the imba ones I saved from FO3.