The OTP functions basically as a USB HID keyboard. So you can plug it into something that is not locked down (like a phone or tablet), and then just copy the code.

The drawback is that the code could be long. A few years ago, the codes were just 6 digits. My latest nano spits out a very long (20 char?) alpha-numeric string.

You can have it behave like a keyboard and simply 'type' the characters of the one time password in. I don't know how locked down your systems but usually keyboards are allowed.

Is it just Mass Storage that'd disabled in USB or the ports themselves?