It's fairly simple, to be honest. Basically all the verses that say "God doesn't experience time the way mortals do", coupled with what we now know about space-time, meaning that space and time are different aspects of the same thing, and the fact that YHWH is eternal and immortal (not bound by time), mean that omnipresence is a foregone conclusion; there is no place where God can't be in a human-finite time, which is equivalent to omnipresence.
I think there are more explicit verses, like "there is no place you can go to hide from God" or "there is no place outside the influence of God" or similar.
A related tangent: all these people who claim to be atheists but think we live in a simulation. If we are in a simulation, then any being or construct which is observing and possibly interacting with said simulation is essentially a god. So I would argue that "simulationists" are de facto theists. A construct running a simulation would have all the characteristics of god: knowledge of everything that happens or will happen (from the standpoint of the simulated; if you can read the simulation logs, you are omniscient; if you save state of the simulation at any point, you can stop and re-start, effectively doing time travel, you can observe everywhere at once, etc.), ability to spawn or de-spawn entities in the simulation, passing through walls, granting supernatural powers, not necessarily bound by the rules of the simulation itself, etc. There's no practical distinction between "the spiritual world" and "the host environment of the simulation."