The short answer is no.
If you're trying to use the Soundcard and the AudioBox 96 for a total of 4 inputs, then you need a bigger audio interface with 4 or more inputs with the same output.
The long answer is here:
On a PC there is no way to synchronize the inputs between the two to be sample accurate using MME, WDM or WASAPI which are all supported by Microsoft, however for real-time audio a custom device driver is needed to bypass windows kernel mixer, this low latency driver is an ASIO device driver. ASIO can only see one interface at a time. Even if your sound card has an ASIO driver, it will not communicate with the AudioBox 96, and vice versa, our ASIO driver doesn't work with your on-board sound card.
To achieve what you want to do, you'll need to an ASIO interface with more than 2 inputs. Like the Studio 6|8, Studio 18|10 or Studio 18|24 or one of the other multi-channel interfaces we offer.
The ASIO device you use will take the place of your onboard sound card and will be able to use Windows drivers for Windows audio playback for things such as YouTube, etc in addition to working as your low latency ASIO interface. Be sure to plug your monitors / desktop speakers into the output of the AudioBox 96.
On a Mac you can aggregate the onboard sound card and the AudioBox 96 as both use Core Audio, and the aggregate device will show up as a single device in Sound Manager and your DAW.