"The dog was limping. It appeared that one of its legs might have been injured."

This means:

The dog was not walking normally (limping),

And it seemed (not certain) that the reason might be an injury to one leg.

So, there is a possibility that the limp is due to injury — it's not confirmed, just suggested by appearance.

let's look at the options:
A. The dog was limping as if it had an injured leg
— This expresses uncertainty or assumption. “As if” means it seemed like the leg was injured, which fits the original. Correct

B. The dog was limping as it had an injured leg
— This implies certainty that the dog did have an injured leg.
Incorrect

C. The dog was limping so it had an injured leg
— Again, this presents the injury as a fact and also implies a reversed cause-effect (limp caused the injury).
Incorrect

D. The dog was limping so as it had an injured leg
— This is not grammatically correct in this context.
Incorrect

E. The dog was limping for it had an injured leg
— "For" here means "because", so this also presents the injury as definite.
Incorrect