Suppose there's two people who found a startup company as associates in the USA. In case one goes temporarily missing, is it possible for the other to take over decisions for both of them (e.g. agree to an acquisition)? Is there some kind of deal stipulated when establishing the society that covers these situations, and is it imaginable that they agreed on terms such as "if Person A can not be contacted successfully for 24 hours, then Person B is allowed full decisional power" (regardless of how poorly thought through such an agreement might be)?
This is absolutely possible. Much depends on the type of business, and the type of decision. I'll assume for purposes of this answer that normally the particular type of business decision requires the assent of both of the people involved, like a decision to sell out the business. I'll also assume that both of the people involved actually own the business. There are several different ways that this could happen.
Most business decisions require the assent of people involved in the business, but not the owners. The corporation's board of directors can often authorize things, for instance. Most boards can meet and vote whether or not all of the directors are present, provided proper notice of the meeting is sent to all of the directors. So if Person A goes missing, Person B can still call a meeting, send out notice, meet, and vote. If Person B's presence at the meeting is enough for quorum, then Person B can still vote. That vote will then be the decision of the full Board, even though Person A was not present.
If the decision requires the assent of the owners, that can usually be handled the same way: notice of a shareholder meeting, attendance by Person B after proper notice to Person A, a vote and it is the action of the shareholders.
It is also possible that such dramatic steps are unnecessary. For instance, say the business is a partnership between Person A and Person B. In that case, Person B can usually speak for the partnership in all affairs (so can Person A). Person B may ultimately have to answer to Person A for what was done, if it violated some agreement or something, but the decision will stick. For many decisions, the law permits a forward agreement that Person A's assent is unnecessary. For instance, in a Limited Partnership, if Person B is the General Partner and Person A is a Limited Partner, most of the decision-making is in Person B's hands. Similarly if Person B is the Manager of a Manager-Managed LLC.
Some decisions absolutely require the assent of Person A. For instance, if there are only two owners and a decision is being made to take the company through bankruptcy, both owners may have to assent. In that circumstance, it is still possible for Person B to go ahead. For instance, if Person A delivered a proxy to Person B in advance, then Person B can vote for Person A. If the proxy was discretionary, then Person B can vote however Person B wants for Person A.
In the most extreme circumstance, Person A can voluntarily make Person B an "attorney-in-fact" by signing and delivering a Power of Attorney to Person B, or else Person B can force the issue by causing Person A to be declared dead or incompetent. Powers of attorney are uncommon, but not so uncommon as you might think - I have seen them used for precisely this circumstance, where one coventurer delivers to the other (or more commonly, each to the other) a limited, durable power of attorney to make certain business decisions in the event of incompetence or unavailability.
Bottom line, in answer to your question it is absolutely possible, and could certainly be done (depending on circumstance) through some kind of advance agreement.