Of course, you can call and ask for anything you like.
But I doubt they are going to be mailing or emailing you off his phone records.
You could potentially successfully get him fired if that is your objective?
But it all sounds like a waste of time to me.
You already know he was texting this woman from his personal phones. Odds are pretty good he's texting her from his work phone now. If you get a record of it, what good is that going to really do you?
He either wants to be married or he does not. I don't see how calling his work, and confirming or not is going to change anything. Either way, you don't trust him right now. Even if he came up clean, you'd still want daily cell records or something.
Oh, I don't blame you at all. I just don't think what you are asking with a worthwhile venture. Divorce him, confront him, or whatever it is you want to do. To call his work and ask for his cell records to me doesn't sound like it's going to do anything but make it all worse.
I wish you the best of luck!
and of course, this is just my opinion.