Illegal weapons export

Generically, this is the "if we don't do X someone else will" claim.

Will they? A claim is being made that may or may not be true. This could be problem with the truth of the claim, not with the reasoning. But if it is true, it is a mere statement of a fact as it is written.

In many cases, there is an implied "so therefore, we should do X." This is where we enter fallacious reasoning. This would be a standard Non Sequitur because the way the argument is presented, we are missing information in the argument to conclude that we "should" do something. Even if we added a premise we would still have problems with the reasoning:

P1. If we don't export weapons to war-fighting countries, other countries will.
P2. If other countries export weapons to war-fighting countries, this would be worse than if we did it.
C. Therefore, we should export weapons to war-fighting countries.

The problem with this argument, is that it has the appearance of a deductive syllogism but it is not, so even if we agree with the first two premises the conclusion could be wrong. For example, what if we export weapons and other countries do as well, and this is the worse possible outcome? If

Only we export = best outcome
Only others export = bad outcome
We export and others export = worst outcome
(remember, we are assuming that "if we don't... then other will" is true, so there is no scenario where no one exports)

Now let's look at your original question:

Sometimes governments that export illegally weapons to war fighting countries argument: if we don't do it, other countries will. Which kind of logically fallacy is this? 

We don't have enough information to say it is fallacious because we are missing the conclusion--we just have a premise.