A sentence starting with “Que” seems to be an equivalent to “May”.
You can find the same construction in Star Wars :

Que la force soit avec toi
May the force be with you

Here:

Que Concorde rapproche les hommes
May Concorde allow people to get closer

The problem here is to translate “rapprocher”…
It litteraly means “make people be closer”. But can either be for geographic distance, or for a relationship.

This motto would translate to something akin to :

May Concorde bring men closer.

It loses a bit of meaning in the translation though : in French, Concorde also designates Concordia, the Roman goddess of agreement and peace.
To elaborate on your sub-questions :

• There is no missing clause to indicate it’s a command or prayer. The sentence is complete and is a perfectly correct way to express a command.
  Que in the beginning of a sentence has two possible meanings :

  • With a proposition in any indicative tense, it’s used for emphasis : “Que la nature est belle !” would translate to “How beautiful Nature is !”.
  • With a proposition in the present subjunctive tense, it is an imperative construction equivalent to “let” or “may” in English. It’s actually the only way to express a command in the third person, or in the first person singular, for which no imperative tense exists.

• “Le Concorde, rapproche les hommes” is not a very natural sentence. It’s correct, but that construction wouldn’t be used. You could say “Le Concorde rapproche les hommes“, which is present indicative : you’re just flatly stating the fact that the Concorde does bring men closer. “Concorde, rapproche les hommes” would be using the imperative second person singular : you’re commanding Concorde or Concordia to bring men closer. “Que Concorde rapproche les hommes” is also a command, or a wish, but you’re talking to someone else than Concorde. These sentences have very different constructions, grammatically speaking ; it just so happens that in all these cases, the correct inflection is rapproche.

• Proche is the equivalent of the English word close and has the same figurative meaning : it can be geographically, or emotionally.

About the matter of deciding which tense is used here, there is no way of differentiating between the present indicative and the present subjunctive form, so nothing tells us if it’s a wish or an exclamation, except that there is no exclamation mark. Meaningwise, though, the exclamative “How well Concorde brings people together !” doesn’t really make sense.