There's some question of what constitutes success in this case, since the first part is a "may". But IMHO there's no question about the arc requirement. If you may lock but don't, it might still be successful. But if you don't even have the chance to lock, I don't see how it can be considered successful to let you move on to after the "then".
I think there is a misunderstanding. The card is not written in a programming language. It is written in English. "If" and "then" do not necessarily belong together.
If you are inside that ship's firing arc, you may acquire a target lock on that ship.
That is a conditional sentence. It starts with the conditional clause if you are inside that ship's firing arc followed by the consequence clause you may acquire a target lock on that ship followed by a full stop. In a programming language you would put "then" at the start of the consequence clause. In English "then" is omitted.
Then you may perform a free boost action.
This sentence is not part of the conditional sentence. It includes one reference to the previous sentence. That reference is the word "then". In this context "then" is about timing. It does not refer to the conditional clause.
Edited by dvor