The Missouri law that allows people to carry guns while intoxicated–or rather, I strongly suspect, the lack of a Missouri law that forbids people from carrying guns while intoxicated*–is not in conflict with any federal law.
*It may even be that a previous Missouri law that prohibited people from carrying guns while intoxicted was recently repealed. But I doubt there’s actually a Missouri law that says “It’s OK to carry guns while you’re drunk!”
However, the Missouri legislature is apparently contemplating a bill to (among other things) “declare federal laws that the state considers infringements on gun rights to be null and void” and subject federal agents enforcing such laws to be subect to state civil and criminal penalties.
Meanwhile, Missouri (like most states) has what’s called “preemption” regarding firearms laws: while local governments often retain some power to regulate things like discharging firearms within city limits (unless actually in self-defense), in most states they may not pass laws regulating the sale, purchase, possession, or carrying of firearms, those things being exclusively the province of the state government, and therefore subject in most states to a uniform statewide legal framework.
So, Kansas City, Missouri, was advised by its city attorney that they needed to ditch their municipal ordinance against carrying a firearm while intoxicated, as it’s in conflict with Missouri’s state preemption of firearms regulations. However, the city council–apparently partly in reaction to the Lege’s grandstanding on nullification of federal laws–narrowly voted not to bring their municipal ordinances in line with state law.
I don’t really think a law against carrying while drunk is a grave injustice, although I do think state preemption with respect to firearms laws is generally a good thing. But even apart from the wisdom of not having a law against carrying a firearm while intoxicated, I can kind of see why the K.C. city council might be pretty cheesed off at the Missouri Lege: If you oppose gun rights, then of course the Missouri Lege’s nullification bill is bad, but I would say even if you support gun rights, these nullification bills (in Missouri and in other states where they’ve cropped up) are just pointless grandstanding that doesn’t really do diddly-squat to actually protect any gun owner’s rights. And, while the relationship between the United States federal government and the state governments is not the same as the relationships between the state governments and their municipalities and local governments, it does smack of hypocrisy for the Missouri Lege to be playing constitutional chicken with the feds while at the same time wanting to uphold their own state constitutional supremacy over the local governments.