This type of Windows-based software does not function with system hardware in either of the methods that would typically be used to run the Windows software - Virtualized Windows, or WINE. This is due to the limitations inherent to each method of software runtimes, and unfortunately is not able to be solved in any safe/secure/sane way.
Virtualized Windows: It won't work for this type of hardware/software integration.
Keyboard backlights are at the system hardware level. VMs do not have the ability to have that level of inter-connectivity with the host hardware directly to interact with this hardware. This rules out using a VM to achieve this.
WINE: This won't work either due to system/hardware control/access restrictions/permissions inherent to user-space (and not admin/system service level) runtime that Wine uses
At the same time, Wine does not have the exposure nor access to hardware resources and modification calls on the access level it needs on the host to handle this type of hardware interoperability. This is because, in a nutshell, accessing the hardware calls is (a) outside the scope of Wine's development, and (b) unable to really be done without running Wine under superuser, which Wine is actively programmed to disallow in recent versions.
So unless Lenovo publishes an Ubuntu version of the software, it is unlikely you will be able to get this software (and therefore the keyboard backlight) working, which will impact the ability for your keyboard to use its backlight. (This being said, there should be a way to have the backlight 'always on' or 'always off', so you might find something like that you can configure from the Windows side to set the default permissions/configuration to have the backlight).
The other possibility is that someone somewhere reverse-engineered a solution that does much of the same. However, this is outside the scope of the original question ("Is there any way you can run a program inside a VM that could access the hardware of the host system itself"), and therefore outside the scope of this answer (you're free to search for such a reverse-engineered solution yourself).