Be they player or NPC, I'll be providing suggestions for equipment and particular training that such an organization is likely to have for ease of generation. For the purposes of this list, I will be considering tech levels from the invention of functional repeating firearms onwards.
Routine gear carried on a day to day basis, when not specifically hunting:
Sturdy coat/vest, such as a motorcycle or riding vest. May be a standard pattern with lots of different customization available to prevent a group of hunters on or off duty from appearing to be related by an organization. Serves to hide equipment and offers a bit of protection.
Some manner of pistol, This pistol will be as state of the art as can be tested to be reliable. It is likely to be chambered for magnum or high power rounds in a caliber that will permit the use of payload rounds containing elements supernatural beings respond negatively to. Typically as large of a capacity as can be carried without excessive size. May have a compact suppressor stored separately if available.
Knives. A standard or above average sized, combat usable folding knife (balisong, tactical folder, switchblade, or similar). Likely to have at least one push knife, dagger, or similar with specialty treatments such as Cold Iron or Silvered.
Concealable light body armor. Run into enough mysterious horrors that supposedly don't exist when you're just trying to go for a walk/have a drink/have a picnic, and you'll decide that at least some armor stays on even off duty.
Pocket Medkit. Life's rough, good to have some bandages.
Tiny Supernatural Response Kit. A minor kit with a few miscellaneous monster repellents, things to prevent critters with infectious bites from turning people, things to seal/turn the undead or help them move on, things of that nature.
Hunt Gear, carried when a target has been identified or is known to be in an area. Includes everything from the routine gear except where things may be switched out for larger versions, such as medkits and supernatural response kits.
Sturdy coat/vest/cover garment to give protection and hide gear.
Compact combat arm. A mare's leg with a folding stock, SMG, or rifle carbine pistol or SBR depending on the tech level. Suppressed if available, loaded with tailored rounds if the target's type is known. In remote areas, full sized weapons may be used.
Larger knife (fixed blade), shortsword, tomahawk, etc. Something capable in melee without excessive size. If the target's type is known, it'll be tailored to dealing with them.
Concealable Combat Armor. A heavier vest (steel shirt, concealable plate carrier, etc), greaves/shin guards, and bracers, possibly with heavy bracer designed to serve as a concealed shield.
Compact medkit. Trouble is expected, more stuff to deal with things going wrong.
Compact Supernatural Response Kit. A larger kit tailored to dealing with the particular target or symptoms reported.
Traps, compact rappelling kit, and miscellaneous other such equipment.
Vehicles: Low profile vehicles with good performance. Motorcycles provide a substantial amount of performance and ability to get a great many places, but leave the rider exposed. Trucks have surprising amounts of power off the line, and a truck capper can make disposal of monstrous corpses a cinch. Vehicles such as hot hatches can handle a chase or getaway, without drawing the undue attention a sports car might.
Weapon skills for gear carried, unarmed combat skills (wrestling/judo in particular, monsters tend to try to grab)
Observation, to keep an eye on the area around them without being noticed.
Stealth and Shadowing, to track targets or potential targets unobtrusively, plan ambushes, etc.
Acting, to serve as bait or pretend not to know or otherwise .
Occult/thaumatology/similar. Knowledge about creatures, magic (possibly the ability to cast spells if magic is available to this group, at least ability to identify magic-derived effects and potentially break/resist them).
Tactics, Traps. Ability to plan ambushes effectively and respond to evolving situations.
First Aid. Self Explanatory.
Fast Draw/Quickdraw. Get gear into a fight, either to surprise the target or in response to an attack.
Above average strength and dexterity for/due to combat training.
Above average resistance to afflictions, death, and mental effects (constitution/health and willpower)
Bravery or fear resistance