This doesn't say anything about what's required in order for the manufacturer to stay in business. Since the premise also doesn't say anything about this, (A) can't make the argument valid. Remember, for a Sufficient Assumption question, if there's a concept in the conclusion that isn't covered as part of the premises ("in order for us to stay in business" is a new concept in the conclusion), then the correct answer, at a minimum, must establish something about that concept. If we pick an answer that doesn't establish something about that new concept, then we have no way to guarantee the conclusion.

This is a sufficiency-necessity confused version of the conclusion. The conclusion asserts that the biggest competitor's staying in business is necessary in order for the author to stay in business ("...our biggest competitor must stay in business"). (B) states that the biggest competitor's staying in business is sufficient for the author to stay in business ("if its biggest competitor stays in business..."). But learning what's sufficient for the author to stay in business does not guarantee what's necessary for it to stay in business.

This is a sufficiency-necessity confused version of the conclusion. In fact, it's expressing the exact same relationship as (B) (just in the contrapositive form)! The conclusion asserts that the biggest competitor's staying in business is necessary in order for the author to stay in business ("...our biggest competitor must stay in business"). (C) states that the biggest competitor's staying in business is sufficient for the author to stay in business. To understand (C) in this way, it helps to recognize that "If author goes out of business, then competitor goes out of business" (what (C) says) means the same thing as "If competitor stays in business, then author stays in business." But, as with (B), learning what's sufficient for the author to stay in business does not guarantee what's necessary for it to stay in business.

This is a sufficiency-necessity confused version of the bridge we want to build. We want to establish "if suppliers go out of business, then the author will out of business." But (D) states "if suppliers do not go out of business, then author will not go out of business." This establishes that the suppliers' staying in business is sufficient for the author's company to survive; but it doesn't establish that the suppliers' staying in business is necessary for the author's company to survive.

If we consider (E) and the premise together, the argument is valid. We know from the premise that if the biggest competitor goes out of business, the suppliers also go out of business. (E) tells us that if the suppliers go out of business, then the author will also go out of business. ("Only if" introduces a necessary condition. So (E) means "author stays in business → suppliers stay in business," which also means "supplies go out of business → author goes out of business.")