Some questions: As you said, you cannot dedicate a complete computer to that project, so can we assume that the solution that you want has to be budget-oriented?
In fact, VGA grabbing with some dedicated hardware can be done with fairly cheap used hardware, like an old PC with a grabber card- as analog stuff is a bit out of fashion for most people, the old grabber cards are available for decent money on ebay etc.
About which amount of data do we talk here? If the datastream and recording duration is not too long, a good value SSD as dumping device allows for less cpu usage and a bit more headroom in a comfortable way to deal with the recordings. After that the dump can be copied to maybe some external HDD to be flexible.
What about VGA to USB grabber cards? Or would the hook up the dedicated PC?
Otherwise:
In filmmaking and movie production, capture devices that usually record from a HDMI input (the output of a standard DSLR etc.) and dump it to Flash (SD-Card or SSD) are standard stuff, including a monitor.
A well known member of that species as the Ninja Atomos
https://www.atomos.com/products/ninja-vThis stuff also would not break the bank, some VGA-to-HDMI converter would need to be used. The HDMI out of the Atomos or similar recorders allows then even for hookup to a bigger screen.