Mount St. Helens began erupting in late 2004 following an 18 year quiescence. Swarms of repeating earthquakes accompanied the extrusion of a mostly solid dacite dome over the next 4 years. In some cases the waveforms from these earthquakes evolved slowly, likely reflecting changes in the properties of the volcano that affect seismic wave propagation. We use coda-wave interferometry to quantify small changes in seismic velocity structure (usually <1%) between two similar earthquakes and employed waveforms from several hundred families of repeating earthquakes together to create a continuous function of velocity change observed at permanent stations operated within 20 km of the volcano. The high rate of earthquakes allowed tracking of velocity changes on an hourly time scale. Changes in velocity were largest near the newly extruding dome and likely related to shallow deformation as magma first worked its way to the surface. We found strong correlation between velocity changes and the inverse of real-time seismic amplitude measurements during the first 3 weeks of activity, suggesting that fluctuations of pressure in the shallow subsurface may have driven both seismicity and velocity changes. Velocity changes during the remainder of the eruption likely result from a complex interplay of multiple effects and are not well explained by any single factor alone, highlighting the need for complementary geophysical data when interpreting velocity changes.