Assuming a very large device was constructed which could harvest up to 20% of the mechanical energy of a tornado. It is a mobile platform of some sort, and one or several of these are ready to deploy in a prone region such as America's tornado alley.
Assume the simplest scheme: the device turns the force of the tornado into rotational work, by deploying some construction of large vanes, kites, or other deflecting surface capable of withstanding the forces. On placing itself in the center of the cyclone, it is able to draw off 20% of the rotational energy and store it before the cyclone wanders away. The construction is unknown at this point. It is designed to handle the most common EF-2 cyclones, and can shut down and secure itself against larger ones. For calculation purposes, it can complete the harvest by maintaining itself within the center of the cyclone for two minutes, then the funnel wanders off with 75% of its original energy (the process leaves 5% lost as deformed metal, heat, deafening acoustic energy, and eroded metal).
While the energy capture device itself may currently be beyond our technology, the question simply concerns a storage mechanism that can recover as much of this huge rotational work as possible, while being accessible to a mobile platform (the unpredictability of tornadoes make it impossible to pre-stage your energy batteries, or the device, within some large construction). As such, the hard science tag only assumes the input of a known quantity of energy at a known rate from an unknown location or time.
What would be the most efficient means to store captured tornado energy given the high rate of delivery, and inability to predict the location?
My worlds are largely wind-powered so this answer can serve several applications generally.