Some studies that helped me with my reasoning: A study has shown that people who are imagining visual stimuli (as opposed to actually seeing it) still show activation in their visual cortex.
A different study that looked at the vase-face illusion found that people's brain activity differed depending on which part of the illusion they were perceiving - if they reported perceiving faces, their fusiform gyrus (the brain's face area) was stimulated, but if they were perceiving a vase, the same area was not lit up. There didn't seem to ever be a time where brain activation showed both patterns indicating perception of both the vase and the faces simultaneously, which suggests that your brain can only perceive one thing or the other.
So my guess at what is happening, in the visual aspect at least, is that your visual cortex would be stimulated, but the stimulation in your visual cortex when you're "seeing" your thought would be different from the pattern of stimulation when you're consciously perceiving what is in front of you.
So for example: you're sitting in front of a striped black and white wall, but you're imaging or "seeing" a solid red colored wall. When you're focusing on (i.e. consciously perceiving) the striped wall, the visual neurons that respond to contrast would be stimulated, but the neurons responsible for color perception would not. When you switch your focus so that you are consciously perceiving the imagined red wall, your color neurons would be stimulated, but your contrast neurons would not.
The prefrontal cortex - the part of the brain responsible for stuff like reasoning, cognition, and other executive functions - probably also plays a role in the thoughts that you "see", because it's implicated to be heavily involved in consciousness.
As for people who can't "see" things in their minds, you're not alone.
There's a condition called aphantasia where people don't see pictures in their head. If I asked the average person how many windows they had in their home, they would probably do a mental walkthrough of their home and count the windows. For somebody with aphantasia, they'll know, but they might not know how they know.
Activation in the visual cortex for the "seeing" part, and likely activation in the prefrontal cortex for the "thought" part.
In the end, not being able to "see" things in your mind is actually a thing - aphantasia.