The Windows input system is designed so an application must have focus to receive normal input, this is not a VoiceAttack-specific limitation.
Unless you are using a device that does not use the standard input system, E.G. a gamepad or joystick, the majority of games would normally stop receiving input if you switch to another application.
Applications that run in exclusive fullscreen mode are also likely to minimize when they lose focus; You could try (borderless) windowed mode instead.
If you don't want to or can't switch focus to the target application, you're going to have to use something other than standard keyboard and mouse input to interface with it. For OBS,
this command line tool (which VoiceAttack can execute using the "Run an Application" action) may be an option.