Most bidets are easily detachable so you can clean underneath. Electronic bidet seats usually have a 'quick release' button that will unlock the bidet from the catch plate so that you can take the bidet seat off.
You can still clean your toilet as normal but make sure not to get any harsh cleaning chemicals on your bidet.
To clean under the bidet, follow the instructions in your user manual to remove the bidet from the toilet.
For models such as the BB-1000, there is a release button that makes it very easy to remove the bidet.
For non-electric models, you will have to loosen the mounting bolts in order to wipe under the bidet frame.
Make sure to wipe off any cleaning solution with water when you clean an area that comes into contact with the bidet.
Always unplug an electronic bidet before removing it for cleaning.