Do hypervelocity stars have Oort clouds? If you had a hypervelocity star travelling through space incredibly quickly, would the hypervelocity star's Oort cloud remain intact? Would the high speed of a hypervelocity star shed the comets and dwarf planets in its Oort cloud due to the star's high speeds, or would the Oort cloud get stretched into a teardrop shape? Could the Oort cloud remain unaffected?
There is no problem with orbiting a star that moves very fast. Remember that space is mostly empty, so there is nothing like air resistance or friction that would tend to deform the orbit.

Stars that become hypervelocity stars likely tend to lose their Oort clouds. They gain their velocity by passing very close to a massive body and getting a velocity increment, for example by having a secondary star or planet ripped away by tidal forces. That already doesn't bode well for the stability of any Oort cloud. 

Simulations show that planets tend to be ejected from hypervelocity systems unless they are very tightly bound, since they have lower mass and respond more strongly to the velocity changes. This suggests that the lightly bound and small bodies in the Oort cloud will be scattered - likely at even higher velocities.