Asking for a large favor

If the PCs get involve in grand events, at some point the player characters are going to need to ask a powerful person for a large favor that goes beyond normal social actions.
While money may change hands, asking for a large favor usually refers to nonmonetary favors. If hypothetically you wanted to buy a magic sword, a sturdy sailing ship, or several hundred head of cattle with an extremely large quantity of coins and trade goods, that is still covered with normal haggling rules.
Maybe you need to convince a barbarian chieftain to pledge his warriors to join a a coalition against a stronger tribe that is swallowing up the smaller tribes one by one by.
Maybe the PCs need to convince the grand magister of a great university to let them read an ancient and highly restricted tome to find the weakness of an ancient infernal monster.
Maybe the PCs need to rescue a dear friend from pirates, but they need to convince the head of the ship builder's guild to loan the PCs a ship and crew.
Maybe the PCs need to convince the Queen to pardon an ally who was falsely accused of a crime (or actually did commit the crime but there are extenuating circumstances).

Setting up the Meeting

Unless the PCs have very high rank and status, they cannot just walk into the royal palace and receive an audience with the king and queen.
While they are generally more accessible than kings and queens, barbarian chieftains, guild masters, coven leaders, high priests, dukes, and other powerful people also generally have safeguards to prevent common riffraff from just barging in and making requests.
Even if the PCs are powerful or sneaky enough to defeat or evade the potentate's security, this doesn't help much. If you are going to ask a powerful person for a favor, you usually want to arrange a meeting through a proper channel rather than using guile or force.
In most cases, the characters have to start at the bottom and work their way up the chain of command. For example, the king normally receives appointments via his seneschal, his chancellor, or his more powerful vassals and knights. Chances are all these high level courtiers of the king have their own assistants and servants.
Sometimes you need to roll Intelligence + Politics to know how the chain of command works, and/or roll Charisma + Investigation to scope out who in an organization is sympathetic to your views (or open to bribes).
At the lower levels you have to bribe, flirt, bully, trick, or charm, or inspire them to make introductions to the higher levels.
Players should try to play to their character's strengths when trying to arrange meetings and game masters should reward good roleplaying and player creativity. Difficulty should be lowered for good role playing or if the players do their homework to find just the right buttons to push to motivate the right people.
One success on such a roll means the NPC is probably going to drag her feet and move the PCs along in a half-hearted manner.
Two successes mean the NPC moves things along reasonably fast but doesn't exactly sell her lord about the trustworthyness and prestige of the PCs.
Three successes on your roll will move the characters up the proper channel, the proper way giving the PCs a polite introduction.
Four successes on your roll means that the NPC will arrange a meeting promptly and put in a good word for the PCs.
Five successes will motivate the NPC to expedite things as much as possible, even breaking protocol or cashing in favors to do it.

Making the Request

Both the player and Game Master need to consider what form a high level meeting takes place. It may be a one-on-one meeting or it maybe with the leader and his/her advisors. In some cases, a public request is made with a large audience which complicates things for both the leader and the PCs making the request.
Once the PC or PCs have an audience with the potentate they want to ask a favor from, the actual roll is Manipulation + Expression, to make the request, though in a formal setting you may have to roll Charisma + Etiquette to get the basic pleasantries and formalities done first. In many cases a bribe gift is given simply for the potentate taking the time to talk with the PCs.
The difficulty is based primarily on how reasonable the request is. But the difficulty can be modified by the status or lack thereof of the PCs and by how well they charmed the lower members of the leader's heirarchy.
For nonroutine social actions of this magnitude, role playing is a must.
Kings, warlords, guild masters, high priestesses and other powerful individuals are still mortal and are thus subject to being intimidated, lied to, seduced, or bribed just like commoners are subject to these things.
That said, powerful people are on guard against being tricked and are often proud and concerned with face so if you want to try to use one of these tactics against them, one has to be subtle to cause unintended. Most high level leaders of any organization have a high Perception + Empathy to smell out cheap tricks.
Which is why the best way to ask powerful people for large favors is to be relatively honest and above board. The player and Game Master do not have to roleplay the negotiation word-for-word (unless you both want to), but they player should least lay out a carrot or stick, probably both.
The carrot is the benefit that the leader will receive if they acquiesce to their request. The stick is what the leader will stand to lose if they refuse the request.
Most leaders have a trusted advisor or Number Two, and they are the real target the PCs need to win over. If you can sway the queen's advisor in a private one-on-one meeting before you see the queen, then the actual meeting with the queen may be very simple and short.
Failure means the PCs are told no (politely). A botch means that the leader is greatly offended by the PCs request.
Three success means the leader agrees to the PC's request. Two successes means the request is mostly granted but there is an additional stipulation or two. One success probably means the PCs are granted something token and mostly symbolic.
Four+ successes means the PCs receive more than they ask for, or at the very least they receive what they ask for and the leader in question feels happy about granting the request rather than reserved and nervous about it.