Any data is sent at or below the speed of light. Solar storms are charged particles (mostly protons) being ejected from the sun and eventually hitting the earth’s magnetic field, causing disruptions in the field (and potentially cool auroras).
Since these storms are just particles traveling from the sun to the earth, they travel slower than light speed, so our distant sensors can warn us in advance at/near the speed of light. This won’t work if the sun were to instantly disappear or change color though, that information would travel at light speed and the probe signals would arrive at the same time.