There are some ways to interact with Mendix widgets. But these are, as far as I know, not documented and not to be considered an official API.
Maybe when Mendix is done with reworking their widget architecture there may be some options for this in the future.
Fabian is correct. We have some ways of including widgets, by including for example
in your widget. Mendix does not support the inclusion of standard Mendix components like this, as it is prone to breaking things and/or changes. R&D is working on complete revising the Mendix client, we might have the abilities then. That's uncertain now.