Note that I have no experience with using any of these products, but I stumbled upon a couple of options:
This VGA to USB capture device. According to the specs it can grab AND convert any VGA input source, and has software that works with Linux. However, at 300 bucks this might be a bit more than you were willing to spend, or you might have even been considering to replace this device with an rPi because of this reason. It does look like an entire out-of-the-box solution for your end-to-end problem though.
There are alternatives as well, such as this device from StarTech priced at $188. This captures HDMI as well as VGA, but I'm not sure of its Linux compatibility.
EDIT: As you correctly mentioned in a comment, this device uses USB3. There is a similar USB2 device from StarTech.com as well, which currently sells for $123 on Amazon. Works with component, composite and HDMI, including pass-through, so you can see the feed on a monitor while the device records the input and delivers it to you via USB 2.0. Unfortunately, I cannot find anything promising regarding Linux support.
Your best bet however is the Hauppage USB-Live2, currently selling for $43. There have been some efforts to get this device to work with Raspbian (see here at Raspberrypi.org) 3 years ago, and you might find that you can get this to work in the current release. The RaspberryPi.org forum post contains quite some details on what to look for. It does require some kernel recompilations though.