What Pokemon would suit this type of character?

I want a Pokemon whose human equivalent would be someone who never attacked close-range, and always attacked from a distance, and they would be like a sniper.

Interesting question. Usually we think of Pokemon as always fighting in fairly close quarters, like in the games and cartoon; it hadn't occurred to me to have a truly sniper-like pokemon strategy.

I think I would pick a powerful psychic type, like an Abra evolution. If they could stay out of the target's sight, they could in theory land a Psychic attack without drawing any notice until it's too late (Depending on how you decide to depict psychic attacks in your particular story. I think they're the hardest to visualize.).

I would agree with that; Psychic types are pretty much often Pokemon that would be inclined to attack from afar.

I suppose you could also look at Pokemon with the actual ability of Sniper; Beedrill or Spinarak's line may for instance be suited to that as well (shooting down people with Twinneedle or somesuch).

I'd like to add Gengar into the mix despite that it has some moves that are close-ranged. Gengar typically attack from a distance with attacks such as Shadow Ball or something similar. Although I'd say Gengar is more like an assassin than a sniper, but hey, snipers are kinda like assassins anyway. Besides, with its Ghost-typing I'd imagine it would be extremely easy for one to escape from any threats.

No one has suggested it? OK, well... Octillery. It doesn't move much, hunts by patiently waiting for prey while suckered onto rocks and mainly attacks using the Beam moves, which imply accuracy at extreme range. I imagine it also camouflages itself (being an octopus), much as snipers do, and to top it all off it has the ability Sniper, indicating that it has a knack for aiming at the enemy's weak points. The Pokédex also states that it instinctively hides in holes, which seems indicative of a Pokémon that knows the value of good cover. I suspect most of its victims probably never notice its presence until it's too late.

F.A.B.

Octillery gets my vote as well. After all, it does learn nothing but "beam" attacks.

Now, if your sniper needs to be small and covert, what about a Roggenrola? It's tiny, just a foot tall, and would rely on fixed-position attacking due to its horrible speed. Imagine this innocent little rock watching you from across a field with its unblinking eye and then - fwip - launching a high-speed pebble into your chest. You'd never see it coming.

Apparently they have incredible hearing. Snap a twig when you enter their line of fire and they'll lacerate you with with bullet-sized ore.

Why not an Aura Sphere? Distance, check. Never misses, check. Pick up a Lucario or a Togekiss, can't go wrong.

a flying type? or better a psychic type?
You know, there are many distance attacks, so I can't think of a pokemon.
Maybe a Lucario or an Alakazam?