I sometimes pay medical bills by phone because there is no online option, and these days they do ask for the CVV.

I also booked a BA ticket that required a telephone booking for the return. The agent mentioned that their system would hold the CVV for one week, and they would have to contact me to get it again if the ticket were not reissued in that time. This also happened last year, they contacted me repeatedly for the CVV, but did not manage to take payment until we checked in at LHR