Several thoughts come to mind.
-Did the lag bolt break only because it was overtightened, or was it a poor fastener? The large home center near me sells fasteners that I will no longer buy. They are too soft, so the heads strip out and, as you experienced here, sometimes they shear off.
-Think of the repair as an escalating series of trials. If you go right to Craig’s core extraction, and it doesn’t work, you have a big gaping core to deal with. On the other hand, if you try to extract the fastener with minimal damage to the wood, you still have various, more aggressive things left to try if it doesn’t work.

If you have a bolt extractor, that is what I would try first. It is almost worth buying one for the (frustrating) experience of trying to use one. On the other hand, if it works, you have a pilot hole to run the same sized bolt into.

If that does not work, or isn’t an option, then you can try to dig out the top as suggested previously to grab with a pair of vice-grips. Again, you may be able to use the same size lag bolt. If you dig too deep, you may just need to get a longer fastener of the same size. If things are too wallowed out, you can go up a size (larger diameter. This may require you to ream out or otherwise increase the diameter of the hole in the vise, which should be soft, cast steel.

One of those should work, but if they fail, then extracting a core as in Craig’s first suggestion starts to look good. The problem with cores and dowels, though, is that you end up (usually) setting your fastener into end grain and, sometimes, the dowel or plug you install can be brittle.

As you suggested, you can also drill a new hole in the vise and place the bolt elsewhere, but I’d rather not do that, if that were my vise, and the other things should work. On my Eclipse vise, there are 4 large lag bolts under the vise going up into the underside of the bench that bear most of the weight and stress. The front holes don’t really support the vise. In fact, are you sure those holes aren’t meant to receive countersunk wood screws as part of attaching a jaw liner? (It has been a long time since I installed mine…..)

It bears repeating: Make sure you don’t buy junky fasteners for critical things. I have no idea if those are good are not, so please do not misunderstand.