There is no way to use distortion correction (or any) plugins in Capture One.
As far as I can tell the distortion correction for Mamiya/Phase One lenses seems to work very well (i.e. it obviously provides correction of complex distortions).
For third party lenses (i.e. all other than Mamiya/P1) there is just a generic distortion correction that only provides linear correction ... which, of course, doesn't help a lot if you have to deal with moustache distortion.
So you would have to use a plugin for Photoshop...