Shoot first, ask questions later - Semicolon or comma

Solution 1:

Ah, the dreaded comma splice. That's what happens when you have two independent clauses, no coordinating conjunction, and a comma—the weakling of punctuation marks—trying to separate them.

The Chicago Manual of Style notes:

Some readers will be distracted by it; some will consider it incorrect; a few will take it as one more sign that civilization is coming to an end. However, as Bryan Garner writes in Garner's Modern American Usage: "Most usage authorities accept comma splices when (1) the clauses are short and closely related, (2) there is no danger of a miscue, and (3) the context is informal."

Professor Tina Blue elaborates:

If the independent clauses are very short, especially if the subject is the same for both clauses, then a comma splice is probably acceptable. . . . When fairly short independent clauses express contrast, a comma splice is often the most effective way to punctuate the sentence. This is especially true if the first clause makes a negative statement, the second an affirmative one, or if the first clause is affirmative, and the second is negative (as in one form of question).

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Your clauses certainly qualify as short, closely related, informal, same-subject, and contrasty:

Shoot first, ask questions later.

And Julius Caesar's been getting away with this since forever:

I came, I saw, I conquered.

But if you're worried that your usage authority will take this as a sign that civilization is coming to an end, you have many other options:

Shoot first—ask questions later.

Shoot first (ask questions later).

Shoot first; ask questions later.

Shoot first, and ask questions later.

Shoot first. Ask questions later.

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Further reading:

Comma Splices and Run-On Sentences

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Solution 2:

The idiom refers to snap-judgment action, quickness of resolve. Of your three punctuation choices: comma, semicolon, or em-dash, the comma exerts the least amount of force towards slowing the reading of the phrase. It results in a cadence/tempo most congruent with the underlying idea.