You can do this in two ways that I can think of. Actually three, since you could do this in the most basic way of all — using standard marking procedures if you're careful, just as dowel holes used to be done back in the old days (and they didn't even have the benefit of through holes).
The second is to drill a small pilot hole into the face piece, position it against the sides of the box (much as when transferring marks from tail board to pin board when dovetailing) and then push the drill bit or a matching scratch awl/ice pick through that pilot hole until it makes a definite mark in the other piece of wood. This is now perfectly in line with the existing hole, you then drill the holes for the threaded inserts as normal using these reference holes.
Whichever of these two options you were using once you have your marks it's vital to have no shift at all during the drilling operation, so workpieces must be firmly clamped up and ideally use a drill press/bench drill for the drilling. This can be done using a hand-held power drill or even a hand drill of some sort if you use a jig of some kind to ensure the drill is held perfectly square.
Third option, no threaded inserts
Fit the front to the three-sided box and clamp up. Drill the front boards and the side boards in a single operation, then drill, ream out, sand or file the holes in the front boards so they just give clearance to the threads on the bolt or screw you'll be using to hold it in place.
Now position the front board and clamp it in place (you're using its holes as guides for the next operation) and thread the holes in the sides using the bolt/screw itself. You can force them in if the threads are not too coarse and the wood isn't too hard, or modify the bolt or screw to be self-tapping if you prefer (the safer option).
In either case it's worth squeezing the wood with a clamp to help prevent it splitting.
Threading made directly in wood by either method can be surprisingly robust and last years of regular use, particularly in harder woods that aren't too brittle. On softer woods the threading can be greatly strengthened by running some thin superglue into the hole after the threads are formed.
*Chuck the rod (or a bolt with the head cut off) into a drill and run it, rotating, against a coarse belt sander or grinder wheel. A sharp point that should be perfectly aligned with the centre line of the threaded rod will be created in a surprisingly short time.