I translate the sentence:
Arzneimittel für Kinder unzugänglich aufbewahren!
as
Keep/store all medicines out of reach of children.
Given it seems to be an imperative statement, why isn't it written instead as:
Bewahren Sie Arzneimittel für Kinder unzugänglich auf!
or could that also be a possibility?
Sometimes instead of using the imperative, the infinitive is used. I think this is so it's more general and not as much like "hey you, do X." Like on a train you might hear
Bitte alle aussteigen! (Instead of bitte steige aus/steigen Sie aus/steigt aus)
or if you're on the Autobahn and part is about to repaired and is in bad shape now, there could be a sign that says
Achtung aufpassen! (Instead of Achtung pass auf/passen Sie auf/passt auf)
Yes, in principle, "Bewahren Sie..." is possible. The other construction (without "Sie") is called an "Ersatzform des Imperativs" (i.e. a replacement form of the imperative) using an infinitive construction.
First of all, this is perfectly good German. Moreover, it often shortens the sentences (see the examples of thekeyofgb or your own favourite examples), which seems to be one reason for its preference over the usual imperative construction.
Furthermore, but this might well be very subjective, the Ersatzform constructions seem more polite to me. Sentences like "Bewahren Sie Arzneimittel unzugänglich auf" are a valid order, however I somehow feel it would be polite to add a "Bitte" somewhere. If I say "Arzneimittel unzugänglich aufbewahren!", then I don't feel the need for a "Bitte" as much as with the other sentence.
You will find that things like that, and especially "Verbote" are often expressed in a non-personal way. This gives the whole thing an official note (i.e. "es ist amtlich").
The alternative is often not so good. Consider:
Rauchen verboten!
This is short and clear. Some authority (usually the state) has spoken. There is no doublt that you have to obey. It is just honest.
The alternative
Bitte rauchen Sie nicht!
does suggest I have still the choice, even if I risk someone disliking me. Or at least, there is room for compromises, discussions, or something. OTOH, for a non-smoker, it is just annoying to be adressed this way (I suppose).
It's basically another way of putting it, in this case using a passive structure. The same is possible in English, of course:
Door to be kept locked shut when not in use!
