It depends on the device.
The classic Osmo iPad Base doesn't have access to the charging port while docked; you will need to take your iPad out in order to charge it.
The new Osmo iPad Base has access to the charging port for all iPad models that are compatible with it.
The Osmo Fire Base has access only for some models. The charging ports are typically on the top edge of the device, so in some cases the Osmo reflector will cover the charging port. Here is a summary by model.
Fire tablet models that allow charging while in an Osmo Fire Base:
Fire HD 10 2017
Fire HD 10 2019
Fire HD 8 2020 (just announced and not officially supported by Osmo yet)
Fire tablet models that don't allow charging while in an Osmo Fire Base:
Fire 7 2017
Fire HD 8 2017
Fire HD 8 2018
Fire 7 2019