I have not come across any official rules with any association that says you have to have certain tack on your horse. After all, barrel racing is not "judged". It is based on how fast your time is only.
You can technically barrel race in any saddle (or no saddle at all) but of course a barrel racing saddle is designed with barrel racing in mind. The seat is deeper to hold you in tight turns. The horn is positioned well to be grabbed turning turning. The saddle itself usually is light-weight for less for the horse to carry.
Most people use a breastcollar to help keep the saddle in place when the horse is blasting out of the turns, but it's not entirely necessary if your saddle fits really well and does not slide back.
Barrel reins are ideal for running the barrels.
You can run in any bit, depending on your horse's needs. You want a bit that is going to give you adequate control of your horse (not in the sense that you are forcing him to behave, but that the advanced bit design optimizes your cues to him), but also have the ability to communicate laterally with him since you will be making direct rein turns. Personally, I prefer browband headstalls with a throatlatch because I think they are more secure than a one-ear.
I think it is a must to have protective legwear on all four legs. I figure that I am asking my horse to make tight turns, so I want to protect him from accidentlaly hitting his leg with another hoof. I personally use the Professional's Choice SMB 3 sport boots on all four legs, along with ballistic bell boots on the front. The brand of sport boot is personal preference, but I would highly recommend to go with a name brand (which means expensive). I have tried the generic boots and they are stiff, do not fit well, and are cheap, which means you get what you pay for. Good brand name sport boots will last for years as long as you take care of them.
∞•*˚ Βгįťţαňγ ˚*•∞
It is not enough to know how to ride; one must know how to fall.