You can't define QED as a strict continuum limit of a "lattice QED" simply because pure QED is inconsistent at extremely short distance scales.
The fine-structure constant "runs" and at energies of the form $\exp(137 C)$ times mass of the electron, where $C$ is a number of order one and $137$ stands for the inverse fine-structure constant, the coupling diverges and prevents one from defining the theory at even higher energies (or shorter distances).
This Landau problem is completely invisible to all orders of the perturbative expansion which is why QED is perturbatively well-defined. In some sense, the Landau pole - the scale where QED becomes inconsistent - is the closest analogy of the "characteristic mass scale" for QED which plays a similar role as the QCD scale for QCD. It's the scale where the coupling becomes of order one and wants to run to infinity (even though QED and QCD make it run in the opposite directions). That's too bad because physics becomes inconsistent at the QED Landau pole scale - a very different situation from asymptotically free QCD.
Lattice QCD itself is pretty demanding - even though one only deals with a rather limit number of sites. The size of the box as well as the size of the lattice spacing are not "spectacularly different" from the QCD scale, or the radius of the proton. However, QED doesn't have any privileged scale of this kind (except for the bad Landau pole scale), so there's no interesting regime or scale at which the lattice QED would be useful, anyway. Note that lattice QCD really becomes powerful to study the effects for which the strong coupling constant surpasses one or so (hadrons and their collisions, for example). That never happens for the QED fine-structure constant. The latter is not confining.
If one used an approximation that circumvents the Landau-pole problem (this approximation would inevitably have to remain silent about all nonperturbative effects because they can't be extracted from QED itself), and if he fine-tuned the UV couplings properly, he could "numerically integrate" QED, at least in principle. All the local Green's functions in QED are perturbatively well-defined which means that one could in principle integrate QED over finite periods of time, too. But because of technical limitations, nothing like that can be done in practice, at least not in any helpful way. The only thing that one could realistically get by this method would be to derive the RG flows and changes of the coupling constant over an order of magnitude or two (at the energy scale): you would substitute the QED Lagrangian at your lattice spacing scale, and derive the effective QED laws at a longer scale - which would be pretty much the same laws but with modified couplings and other parameters. And these things can be derived theoretically, without any help from computers, too.
I think that it's fair to say that only QCD and its equally confining cousins can be - and has been - usefully studied by these numerical techniques. And even in that case, one still deals with the arguably eternal problems of QCD - problems with fermion doubling and other fermion problems; problems to naturally impose all of supersymmetry, and so on.
