Trying to capture analog video such as VGA and transmit it over Ethernet is going to be technically challenging, and of questionable wisdom (at least without compression). A conventional ATmega-based Arduino is just about the worst contemporary platform you could choose for this job - even aside from the VGA issue, it is a poor choice for Ethernet requiring an expensive add-on. But just for the sake of "can it be done" ....
If you can threshold your signal to a digital level (assuming a high contrast image) this probably can be done, crudely, but will be quite tricky. The display is likely to be very blurred horizontally - really, it would only be worth doing for bragging points.
The first thing to realize is that the standard VGA 640x480x60 mode has a pixel clock of 25.175 MHz, and most others will be even faster.
One approach would be to try to write a program which samples a digital pin (directly, no Arduino abstractions) to memory as fast as possible. This would not capture every pixel, but rather something like every 10th. You would then have to send these over the network before you fill up your available memory.
The tricky part is that you then have to go back, and scan another frame of video with a slight offset, so that by getting every 10th pixel again, you effectively get the pixel adjacent to the first. And then you do it again, and again, until you have collectively transmitted the entire screen, which hopefully hasn't changed much as you did it.
It's going to be tricky as you will have to work your delays from the sync signals, and you really want a delay which is a fraction of ATmega's clock - which won't happen without external circuitry (programmable delay buffer, etc)
The whole project would be a lot more feasible on an 80+ MHz ARM Cortex part, as you could then use a counter/timer to generate pixel-or-less-adjustable starting delays relative to the sync pulse. An FPGA would likely be an even better choice.