OK, here's my own answer.
There are basically three options to work with:
- Delete the electromagnetic field, and replace it with something completely different.
- Delete the electromagnetic field, and try to adjust the other forces to make up for it in a way that doesn't re-introduce another kind of light (e.g., massless pions).
- Alter the electromagnetic field so free photons can't propagate.
I have spent some time contemplating options 1 and 2, and they always seem to go off the rails rather quickly, becoming sufficiently complicated that you might as well just build a whole new universe from scratch. El duderino's answer is in the realm of option 3, and so is my next alternative:
In order to restrict photon propagation, rather than altering the properties of space as in el duderino's answer, we can instead make the photon decay over relatively short distances. In order for photons to decay, they will need to experience time, which means they need mass--but not necessarily very much mass. Giving photons mass restricts the range of the static EM force as well, replacing the usual Coulomb potential with a generalized Yukawa potential with exponential decay, just like the residual nuclear force carried by massive pions
$U(r) = -g^2\frac{e^{-\alpha m r}}{r}$
where $g$ is the force coupling constant, $\alpha$ is a scaling factor to get the units right, and $m$ is the mass of the gauge boson and determines the field decay rate. As $m$ approaches zero, this simplifies to the normal Coulomb potential, so if $m$ merely remains very small, we can retain atomic and chemical structure with only minor perturbations, and possibly even retain some macroscopic EM effects (or not, as we choose).
However, giving photons mass is only a necessary, not sufficient, condition for short-range decay of free photons. We also need something for them to decay into. The obvious first option to try is just having photons decay into electron-positron pairs, so the universe ends up filled with a positronium gas as the dominant form of matter, rather than hydrogen. That might be a cool avenue to explore independently, but it means that the photon mass must be at least 1.022MeV--which in turn limits its range to nuclear scales. That would definitely screw up chemistry! Next we might be tempted to ask if photons could decay into neutrinos--and unfortunately, the answer is "no". The reason is that neutrinos have no electric charge--only weak isospin--and so do not couple to the electromagnetic field, and cannot participate in production, decay, or absorption of the electromagnetic boson.
So, we're going to have to add a whole new fermion field to permit a route for photon decay. Fortunately, considered and rejected both electron decays and neutrino decays as suitable candidates, there is a glaring whole in our universe's particle inventory which nicely solves the problem!
Note that electrons have both electric charge and weak isospin, which is why they interact both electromagnetically and weakly. Neutrinos have only weak isospin, which is why they only interact weakly. But where are the particles with only charge, and no weak isospin?
If we add such particles and give them very low mass, that solves most of our problems. Being low mass, they will not bind to atoms, so electrons are left unmolested to continue supporting chemistry. Having charge, they will support photon decay...
...but, we may have a new problem: we may have just destabilized the electron. If there is a low-mass weak isospin particle (the neutrino), and a low-mass charged particle (the new thing--let's call it an electrino), then perhaps electrons can decay into a pair of electron-neutrino and electrino, splitting up its charges between them.
Fortunately, though, two other conservation laws come to the rescue: conservation of spin (i.e., angular momentum), and conservation of lepton number! If "electronness" is also conserved (as it appears to be in the low-energy regime, modulo whatever process ends up explaining the matter/antimatter asymmetry), then an electron could produce either an electron-neutrino, or an electrino, but not both. And giving our new particles half-integer spin (i.e., making them fermions) just like other leptons means that a parent particle must be either spin 0 or spin 1--which an electron isn't.
The introduction of a new low-mass charged field will change how vacuum polarization works, and introduce an additional weakening of the electromagnetic field over very large distances--but that will be difficult to disentangle from the exponential decay induced by the photon mass, anyway.
At this point, ensuring that photons are not useful for vision is just a matter of tweaking the mass ratios of the photon and electrino to keep the photon half-life really low, thus limiting its range. Unfortunately, the mathematics governing boson decay rates of this type (the closest analogues would be the decays of W and Z bosons) are way over my head, so I have to get a little bit handwavy with the exact values--but thus far, it seems like this approach of simply adding a decay path for photons should work.