The contents of this post (pretty much like any other) could get you fired if you follow them incorrectly (or, well, correctly, depending). The following is meant to be informational, to help you think outside the box, but please use your own discretion so you keep your job.
As an employee, we've all been in that situation. You know, the one where you've got a deadline hanging over your head, but a decision has to be made by person or persons who are not you. Realistically, no one can blame you if you miss your deadline, but in reality, we all know that people do blame us for things outside of our realm of control.
I'm a huge fan of getting consensus the right way; talking to people individually or in small groups and getting agreement, then sending out an email letting everyone know they've agreed to it...nice, efficient, and people think well of you when you're done. But if the deadline is this afternoon and you got the problem this morning, elegance and simplicity are not really in your game plans for the day.
Now, if the parties in question that have to make the decision don't know (but you do), you can go the beg forgiveness route. In this method, you send an email to everyone telling them the decision you are going with, the time frame by which that decision has to be implemented, and an explanation that if any of the approvers don't wish to approve, then they will be responsible for missing the deadline. Always make the time frame for approval of the decision a few hours earlier than the actual deadline; people will email 10 minutes after, and if its a Vice President, you are going to have to find a way to put the bullet back in the gun if you've already fired.
That's a fast and dirty method of getting approval--everyone's on the thread, and everyone is aware that they will be responsible if they have objections. Now, some people, like bulls in china shops, won't care. Porcelain will fly, but the project will not. Some people, however, will send a few emails to ask questions to make sure they at least appear to have done their due diligence. The majority of people? They won't respond at all during the time frame. You might get a no vote tomorrow, or when they get back from vacation. Usually though, you'll get silence if they agree, or if they don't care enough about the issue to respond.
I do not advocate letting them know about their choices at the last minute, unless that's when you found out. Deliberately reducing the time window can bite you in the butt. But doing it legitimately can get those decisions moving, and allow plenty of credibility when you're begging forgiveness for any decisions made.
Got an approver who is always in meetings and rarely answers email? Got an issue you need to move forward with? Send an email to them and let them know that you are doing X unless they tell you otherwise by Y time. Then follow through. When they eventually find the email, you may have to beg forgiveness. But you did consult the approver, to the letter of the law of approvals (even if not especially the intent).
In a meeting with the rep who is about to fly back to a far corner of the earth and no way to reach your approver? Make a decision on the spot that reserves your options if you can. Don't sign any contracts or agree to any financial clauses (at least substantial ones), but if the choice is "yes or no" and answering "no" means the opportunity could be lost or hard to get into the queue again, answer "yes." Follow up with your approver pronto and get your word backed by their law. You won't always be right, but they will appreciate you keeping their options open.
Note, these things can get you in big trouble (maybe even sometimes fired). If you have an option to talk to someone or use these alternate routes, you should probably always talk to them. I do say "probably." Some things--deadlines, projects, friendships, companies--are worth risking everything for.
I leave it up to you to decide what you're willing to risk.