Unless your game is the Volunteer State! As of last year it is completely legal here in Tennessee to bring firearms into bars and, well, any other establishment that does not house our politicians. Including concealed carry. Currently, owners can choose to ban weapons on their premises, but there is a push in our legislature to do away with the right of a property owner to prohibit weapons on his property. Not sure if it will go through or not. Best bet for anyone running a game with a gun-bunny character that wants to keep it real is to go online and look up your state and municipal firearm laws. They can vary wildly from state to state. Or even among cities/counties in the same state when you get down to certain particulars like storing them in a place of business or discharging the weapon. Or even in how the weapon is classified. A guy in Knoxville made this AK-47 conversion that shortened the barrel, removed the stock, removed the brackets such that it could not use a sling, kept the 30 round banana clip and took it to the local park wearing camo and tactical gear. He then proceeded to stalk through the park and, in the process, scared all the moms there with their kids to death and created a general panic. He ended up being detained by the police but, due to the conversions he had made on his AK-47, it now fell into the pistol category. Therefore, it was completely legal for him to carry it into any state and city parks. He is now suing the state for wrongful detention and harassment.
For an amusing aside, I recently did that research for my little town here in Tennessee in regards to dealing with a nasty groundhog problem and discovered that my city has a long list of prohibited projectiles that includes; I crap you not, and I quote, "Snowballs thrown with malicious intent." No mention of arrows though in that page long list...
In 90% of states, though, you would be in deep bantha poodoo for having an armory under your bar.
Looking at the links provided (thanks UmbraLux; I don't hunt so I hadn't seen the non-shotgun sabots) if you have the money you can buy as many sabot round kits as you can afford. The Remington 12 gauge sabot shells are around 22 dollars for a box of 5 according to the internets. I doubt the ATF is really tracking those much. At least no more than they do normal ammunition purchases. Which, they do track and follow up on someone buying bulk ammo. Congress and certain state legislatures are currently throwing around bills that would ban the ATF and other agencies from doing that; but as of right now, they do. Largely as part of efforts to head off smuggling down to Mexico and the cartels. You can still buy pallets of all the ammo you want! All nice and legal. It just means you will be getting looked at.
FYI, Section 8 Discharges are no longer used by the military. Army Regulation 635-200, Active Duty Enlisted Administrative Separations. Chapter 5, paragraph 13 (thanks Dad!) now covers discharges for personnel with medically diagnosed psychological conditions. AR 635-200 took effect back in 2005. So, it would depend on the character's history and how much of a stickler you wish to be. If they were drumming him out for having reported vampires and thought he was PTSD or such, he would get a Involuntary Chapter 5-13. That is if they really wanted him out and pressed it.
Those dishonorably discharged from the United States Military and those having been adjudicated as mental defectives or incompetents; both of which this character could fall into; are prohibited by the Gun Control Act of 1968 for owning a firearm. And, yes, this means anyone drummed out the military for being a homosexual can never own a fire arm ever again.
This makes life hard for this character. Very hard. It means that anything he is purchasing and procuring is either black-market, purchased from a dealer not doing his proper checks, from personal sales (which are not monitored) using the gun-show loophole. But, the moment he shows up on the ATF's radar, he's in for a full on federal warrant search and seizure visit. And these guys are really, really good at finding hidden gun vaults and the like.
NOW, if his command did not hate him and he had some friends in the chain; they would have told him what the bureaucratic tea leaves were reading for his future in the military and it would not have been hard to get him a voluntary discharge for a wide range of reasons. Which would allow him to legally own firearms.
SABOT ROUNDS. The purpose of a saboted round is to allow one to fire a smaller projectile with the energy normally reserved for larger rounds. Such as putting a 7.62mm tungsten alloyed armor piercing round inside a .50 cal sabot so you can fire it out of your .50 cal weapon with all the energy of that .50 round behind it. This is useful in armor piercing applications mostly. Or, in the civilian hunting arena, as I have learned, to be able to fire smaller rounds without needing to carry a different weapon. Shotguns use it to gain a great deal of additional accuracy and energy over range. A DU round would use a sabot round to keep the DU core from tearing your barrel to crap.
EXPLOSIVE ROUNDS. A lot of the time, people will refer to incendiary tracer rounds as "explosive" rounds. This occurred often in accounts of their use on the Eastern Front of World War II. Especially from accounts of snipers on both sides. There were actually explosive rounds and still are. They are utterly prohibited by all sorts of various international treaties such as the Hague and Geneva Conventions from being used against human/non-materiel targets. They were used on the Eastern Front of WWII because Lenin removed the USSR from all previous "Tsarist" international treaties and the Germans and Soviets really hated each other at that point. All German snipers were strictly verboten from taking any explosive/incendiary rounds with them if they were transferred to other theaters.
True explosive rounds are designed for armor piercing; which is legal under all those treaties. They work not by some fancy, miniature fuse system in the round itself. They have a magnesium or phosphorus tip that ignites upon impact due to the KE of the impact which, then, detonates the explosive core behind the tip. There are many, many variations of exactly how this works. Most of them are designed to work in 20mm and larger rounds. Largely for aircraft, attack helicopters, and AFVs. They do make .50cal explosive rounds for use on armored targets. The problem is, if you shoot a soft target with them (say a human), the bullet will punch right through them and not detonate as nothing will create enough frictional heat to ignite the incendiary. Not even body armor would be enough to set it off. Maybe! Maybe if you hit the strike plate of the really heavy duty military issue body armor. Bigger rounds can and do use mechanical fuses as well, but getting one to work in even a .50 cal round would be tough. Anything smaller is nearing impractical and/or extremely expensive.
As for doing more damage. It's blow through. You hit a human target with a .50 cal round inside effective range, do you really think that having it go boom will make any bit of difference to anyone but the forensics team that shows up later?
Having HE rounds used in a firefight in the US is a good way to have every federal law enforcement agency and Homeland Security crawling over your city like ants in a kicked over hive. No one with any brains is going to see enough effectiveness out of their use to risk the attention it will bring.