TL;DR: this is a dangerous and wasteful way to make a gun. Probably not worth it, as almost any other kind of weapon will be better.
is it possible that shooting antimatter at someone would just vaporize them?
Well, yes. They may indeed be vapourised. The problem is that the "vapour" that results will be a cloud of very hot, very dense plasma, which will expand rapidly until it reaches equilibrium with the surrounding atmosphere. Also commonly known as an explosion. Delivering enough energy to vapourise a human body will create a very large explosion.
Whilst some people are of the opinion that there's no such thing as overkill, there is such a thing as efficiency... you can kill a person, or disable a vehicle, by poking a hole through it, which requires a lot less power than reducing them to plasma and causes a lot less collateral damage.
Speaking of collateral damage, the ultimate end result of annihilation is the production of a lot of gamma rays. These have a short free path in meat and metal, but can travel quite a long way in air. They're a serious risk to bystanders and indeed the wielder of an antimatter gun who is stupid enough to use it at short range. The annihilation flash will also be extremely, blindingly bright.
(also note that if you shoot enough antimatter, everyone and everything will be annihilated... effective yield is 43 kilotonnes of TNT equivalent per gram of antimatter. Use sparingly)
And if so, is there a way to fire antimatter through the air without it annihilating as soon as it touches the air?
You need to confine it in something... electromagnetic confinement in a Penning trap is what people try and do currently. You might be able to miniaturise the traps so that they're merely grenade or bullet sized in stead of bomb or vehicle sized. Other people have handwaved antiparticle confinement inside fullerene molecules... if this involved some kind of confinement other than electrostatic then it might work, but I can't find any serious analysis of this sort of thing so it probably won't work.
Remember that if your antimatter becomes deconfined before it hits the target, you'll end up having a really bad day. If your confinement systems run out of power, or they're badly damaged, or burnt, they'll go boom. If you're subject to highly penetrating radiation, that could give some of your antimatter enough energy to escape confinement which might then damage it enough to deconfine the rest, too. If people are throwing around antimatter weapons, then there will be a lot of gamma rays flying around, and these are indeed highly penetrating radiation.
Basically, the end results would be energy cannons... that fire orange beams that are composed of a mixture of most types of antimatter so they'd annihilate anything they hit except for what the air is composed of.
No orange beams for you, sadly. In an atmosphere, you wouldn't be able to make a pure enough vacuum over a long enough distance and maintain it for sufficient time to throw antiparticles down the middle of the beam to zap the target, and if you could you could use the beam-generation system as a weapon all by itself without all that dangerous mucking about with antimatter.
In a vacuum you could fire an antiparticle beam, but particle beams suffer from thermal blooming that strongly limit their range compared to pretty much any other weapon system. You can increase particle speed to improve range, but once you get over beam energies of about 1 GeV per nucleon you may as well just use regular matter instead (because the kinetic energy of the particle starts significantly exceeding the energy released by its annihilation).
You'd be better off putting your antimatter in a missile warhead or cannon round, or using it in your own generators or rocket engines.
Do note that there's a common canard that if you bash a blob of antimatter and a blob of matter together, they'll just blast themselves apart. This is somewhat unlikely: annihilation releases a spray of highly energetic and fairly penetrating radiation. Both blobs will be thoroughly toasted by this radiation to quite a depth, and that means they'll explode from the inside, not just at the contact point. You'll get two more-or-less spherically expanding clouds of plasma which will interpenetrate and annihilate as they expand. A small quantity of antimatter may escape in a vacuum, from the back face of the antimatter blob. There's no danger of getting hemispherically expanding clouds that won't contact; physics doesn't work that way.
You can of course trivially fix this by having the mass of the confinement and delivery system exceed the mass of the antimatter warhead, and having a decent amount of that mass being behind the warhead. For really big lumps of antimatter, break them down into smaller, separate confinement devices. There's little danger any antiparticles will escape.