The Zat-gun fullfills very usefull jobs in the Stargate SG1 series, allowing stunning and providing a means to overload (and disable) equipment. I would have thought they would have brought some to both Atlantis and to the mission ending up on the Destiny. However, this is (as far as I know) not the case. I also don't recall for them even being mentioned in either spin-off. Hence the question, is an in universe explanation given for not having Zat-guns on Stargate Atlantis and Stargate Universe?
Word of God answer:
Joseph Mallozzi, Screenwriter and Executive Producer for both Stargate: SG1 and Stargate: Atlantis explicitly answered the question of why he didn't include Zats and Goa'uld weaponry when he wrote the scripts for Atlantis in a Q+A on his blog. Evidently the crew of the Atlantis simply decided that the weapons weren't what they liked or were used to.
Max writes: “I have been wonder[ing] for over 5 years why the Jaffa weapons (staff and the handheld zapper) weren't used in the Atlantis series.”
JM: Staff weapons and zat guns are unwieldly. I’d think military personnel would be more comfortable using weaponry they’d been trained to use.
Out of universe answer:
Clearly the intent with both SGA and SGU was to create a unique 'look' for each series. Having substantial amounts of Goa'uld weaponry (when the goa'uld barely get a mention in either series and each use of their guns would require an extra line of exposition to explain where they come from) makes little sense from a screenwriting perspective.
For the record, there is at least one occasion in Stargate Atlantis where zats are used; by Teal'c in the episode "Midway".
It's possible (though unlikely) that the SGC has a limited supply, and couldn't risk sending them on an extended (possibly permanent) trip. I find it hard to believe myself, since they should have plenty of ways to acquire new ones.
However, at that point in the series (Season 7-8 of SG-1) there was a diminished usage of Goa'uld tech - it was around this point that Teal'c dropped his staff in favor of a Tau'ri weapon.
It's also worth considering that the forces sent to Atlantis had to be ready to become self-sufficient if needed, or at least capable of functioning without a high technological base to work from ('Atlantis' could have been a city filled with broken machines and dust, instead of what they found). Many, if not most, special forces troops (from which SGC teams are often drawn) are able to make gunpowder if needed, and every soldier knows the basic maintenance of his weapon.
The same can't be said of Zats, which seem to require some unique materials.
In relation to what Jeff said, I doubt that the SGC would risk sending Zat's on what could possibly be a one-way trip, especially when those weapons were needed to arm the recently formed Free Jaffa nation.
Plus, shortly after arriving in the Pegasus galaxy, the Atlantis Expedition acquired Wraith stunners, so I doubt that Sheppard or the SGC would bother sending Zat's anymore.