To be able to surf/survive a tornado, you must be able to have the consistency of tank but light enough for a human to carry.
I'm going to assume this extreme sport always gets you in trouble or puts the surfer in the worst case scenario.
Here are the essentials every tornado surfer will have. First and most important every surfer needs a parachute. Of course this can't just be any parachute since normal material can be shredded by debris flying at over 200 MPH. That's why this parachute is made and engineered to work using Cuban Fiber, a lightweight fiber that's stronger than steel. Of course the bag concealing will also be made of the same material. (This could also be used to be "wings" on the suite like in @Ahriman answer).
Now of course you want something to protect your body from all this debris also! Lucking there is a new class of magnesium-alloy that is one of the strongest metals ever produced that is less dense than water! So this lightweight solution of body armor will be perfect to stack on.
In this suite will have to be small oxygen tanks like in @Ahriman answer. These will be small enough to just last the duration of the ride, nothing huge like when scuba diving. The surfer can't get suffocated from the wind! (try putting your head out the window going down the highway, even that's tough to breath!)
Of course under the armor would need to be some extreme cushioning like Sorbothane, which can help protect from fire and such also, and probably a neck brace and sorts like that for impacts. (Human bodies are too squishy).
Possible facts - Anything over 12m/s is pretty dangerous when falling. The sudden stop is what kills. So here are some techniques to survive a fall.
Scenario 1 (Launched High & possibly far)
This is the best scenario a tornado surfer would want to be in. This gives the person plenty of time to react to what is going and will allow them to use their parachute to float down safely. Assuming projectiles don't come from the tornado and snag the parachute, it will become an easy ride. If the parachute does get grabbed, the wings on the suite can slow the descent hopefully to safe speed.
Scenario 2 (Launched low & and fast)
Coming out of the bottom part of the tornado, launched like a human missile, only hopes and prayers will help you here. The surfer is now faced with great danger, tons of things to collide with. Only the protective gear will save them now. Hoping the body armor can withstand colliding or being pushed through walls and cars or slammed into metal structures. And the cushioning underneath can absorb the impact!
Scenario 3 (Launched midrange)
Any experienced good surfer will be able to react using their body suit to glide out and possible up depending the wind. From there they can get their parachute deployed to slow their decent. Even though this would be a rough landing, they will survive to surf again!
Scenario 4 (Anything slow)
High up, mid range is no problem for these guys at low speed. It's like surfing a 4 foot wave in the ocean for professional water surfers. Low range is where it still can be dangerous, but with the proper gear and the perfect falling techniques, these guys have come into much worse!
All in all this is not a sport for the fearful, also not a sport for the unhealthy. This is a high class high energy, highly dangerous sport that only few and brave and (stupid?) every attempt.