According to writer/director Tom McLoughlin, Jason Voorhees was originally supposed to make use of an uzi submachine gun in Friday the 13th Part 6. As any horror fan knows, the killer in the original Friday the 13th film was Pamela Voorhees, Jason's mom. This left the inexplicably alive Jason to carry on the family legacy from Friday the 13th Part 2 onward, and one can't say he hasn't done it with gusto. Jason certainly isn't the most talkative guy, but he clearly enjoys his job, having dispatched dozens of unfortunate victims throughout the 10 Friday the 13th installments he's starred in so far.
As the series progressed, Jason also got a lot more creative with his methods of murder. That makes perfect sense, as the main draw of a slasher movie is seeing just who will get offed and how it will happen. Jason graduated from using standard weapons like a machete and an axe, to such implements as gardening tools, an electric guitar, toxic waste, and even a party horn. Per McLoughlin though, all those kills might have been outdone by the sight of Jason straight-up shooting people in the franchise's sixth outing.
During a recent interview with the Midnight's Edge show on Youtube (via Arrow in the Head), McLoughlin talked in depth about 1986's Friday the 13th Part 6: Jason Lives, likely his most famous work as a filmmaker. Jason Lives is generally quite beloved by fans, as after Friday the 13th Part 5: A New Beginning decided to replace Jason with a copycat killer, McLoughlin opted to resurrect the slasher icon as a super-powered zombie. Jason Lives also introduced a playful humor to the franchise, clearly not taking things too seriously. To that end, McLoughlin revealed that his original script called for Jason to blow away a few victims with an uzi. Here's his full quote:
I turned that in (a 15 to 20 page treatment for JASON LIVES), and it's pretty much verbatim exactly what the movie is, in terms of the characters, the scenes. The only thing that is different that I wrote and then realized, 'No, this is not a good idea', was when Jason kills the paintballer that has all the weapons and stuff that he takes, you know with the utility belt and the machete and all that stuff. Originally these were not paintballers, they were hunters out there, and this guy was Mr. Macho with all that stuff and he had an Uzi. So when Jason ripped his arm off and gets the machete, the next time we see him, in front of the motorhome, he's got an Uzi strapped to him. And I thought from a visual standpoint, it's like 'Holy shit, Jason Rambo!' And so when he goes into the motorhome and he bursts out of that bathroom on the girl, he just opens fire and shoots the crap out of her and shoots the crap out Cort behind the wheel and then the motorhome crashes. But I went, 'You know what? Anybody can do that.' ... If I'm bringing Jason back from the grave with this superhuman strength, that he can turn a head and rip it off and punch a heart out and decapitate three people with one swoop, this is just not right, and it's also an imitatable act, and I did not want that."
In a movie where Jason does things like decapitate three people at once, literally snap a man in half with his bare hands, and rip a victim's heart right out of his chest, it's hard to say that the image of Camp Crystal Lake's resident monster gunning down folks with a machine gun would have seemed that over the top. Still, it's not hard to understand McLoughlin's reasoning for removing the scene. The other crazy kills mentioned above are all acts a real-life murderer couldn't imitate, and could only conceivably be carried out by a supernatural entity, making them uniquely creative. Picking up a gun and shooting people is something anyone can do, as evidenced by the disturbing amount of mass shootings taking place in the U.S.
Aside from that, using a gun wouldn't really fit in with Jason's usual modus operandi. While Jason made use of a speargun in Friday the 13th Part 3, that's still a long way from using an actual firearm. The vast majority of Jason's kills are of the up close and personal variety, presumably because Jason likes to watch his victims squirm. Shooting people would seem too detached for Jason, and contrary to his established characterization. Well, his established characterization when not turned into a body-hopping demon, a cyborg, or back into a little boy. Yeah, these movies are strange. Still, that doesn't stop fans from loving them.