8 bit PC integrated with VGA video card (similar to ben eater project)
Using an EEPROM for the video memory

.. why? (Is this a self-imposed restriction, a parts availability issue, or part of some grander scheme to achieve some other purpose in the design?)
As you've spotted, you can't freeze the image; you have to continously generate the VGA signal. You could take a copy of the image in RAM, but presumably there is a reason why your design is allergic to RAM?
The normal solution to using single-port memory without corrupting the displayed image is to only do writing in the vertical blanking interval. That does limit your write bandwidth, but depending on how many pixels are on the screen it may be sufficient.
An intermediate solution might be to have a small amount of video RAM which is character-mapped rather than pixel-mapped, and use the EEPROM for character generation. Same VBLANK suggestion applies.