Extender HDMI over Ethernet

Anyone of these HDMI extenders (excluding the wireless ones, in your case), will reliably do the job for you - even the low cost ones, as your distance is fairly short. Some stand-outs to look at:
- AdderLink Dv100: Extremely high-end, with small form factor and doesn't require a power supply on either end.
- KD-X200PoHK: Supports Ultra HD (4K) resolution and requires a power supply only at the source (not by the screen).

To convert the VGA source to HDMI (before extending it) - look at the VC180 or the CVH-01.

You may also want to consider deploying a 2-port HDMI switch at the source. This will eliminate the need to disconnect and reconnect the converter to the extender. You could have the converter permanently connected to port #1, and have the extender permanently connected to the switch's output. When the source happens to be VGA, you connect it to the converter, select port #1 - and you're good to go. When the source happens to be HDMI, you connect it to port #2, select port #2 - and you're good to go. Anyone of these 2-port HDMI switches will accomplish this.

Note: If you select an extender that supports 4K - make sure the switch also supports 4K.

 

Even more compelling is this new offering in an all-in-one solution.

This HDMI/VGA AV Extender Kit accepts 1 VGA input and 2 HDMI inputs, allowing you to select your input or automatically select the most recent connected source. If the source is VGA - the unit converts the analog signal into digital HDMI (including audio).

Another advance version up-scales the VGA to 1080p HDMI, and also de-embeds the audio from the HDMI into a separate output.