There are THREE kinds of DVI connections: DVI-A (Analog ONLY, RGB+Horiz+Vert Sync, compatible with VGA), DVI-D (Digital ONLY, compatible with HDMI) and DVI-I (Both Analog & Digital) which can interface directly to either VGA or HDMI Monitors, with the appropriate adapter cable:
Your diagram is puzzling.....is the originating HDMI interface for some unspecified reason MISSING the Digital Audio data stream??? Or are you intentionally trying to eliminate the HDMI handshake that requests either STEREO or DD5.1 data stream???
When you say "RGB", it MEANS you are trying to interface a device that has a DVI-A (Analog ONLY) interface, which are very rare except in PCs, dating back to early PC DVI-A to VGA DB-15 Monitor interfaces (or just the Analog portion of the later dual purpose DVI-I interface). This would require a three channel video sampler feeding a transcoder and Audio/Video mulltipler to bring in the Digital Audio data stream. I don't believe that any of the above devices do this functionality. Following "VGA and Audio to HDMI SCALER" appear to do what you requested, using a DVI-A to VGA Adapter cable on the input:
If you instead have the much more common DVI-D interface (probably with HDCP copy protection), as found on early HD and HDTV equipments, then both audio & video input signals are already digital and "only" have to multiplexed together into HDMI format, and the Monoprice 5369 mentioned above "should" work:
The above Monoprice 8146 Multiple HDMI/DVI/C-V Input Switch with HDMI Output only accepts DIGITAL DVI-D input, so it doesn't support DVI-A interfaces....and although the Component Video Interfaces are frequently called "RGB", in fact they use an entirely different color space: YPbPr. Since Digital Audio is already embedded within HDMI, why would anyone expect to see a Digital Audio input connector next to each HDMI Input???? I see no reason it wouldn't do the same job as the Monoprice 5369, plus additional switched inputs.