If your source is local, and you record local, then you don't need an Internet connection at all.
Typically a recording is done locally (i local to PC disk drive) so no, that wouldn't use up ISP data allocation
HOWEVER, you have to think this through
Are you saving your recordings to a drive folder that you have set for cloud sync (ie nothing to do with OBS)... then of course it would consume data allocation.
OBS Studio is NOT a SaaS app, that has to utilize Internet data... its just a common use case to livestream using OBS which of course does use the Internet
But, lets say you are gaming, using plugins that are Internet based, etc - then of course you are still using bandwidth (not for recordings themselves, but for your general activity)
I hope that helps