Although, the rest of the party suspects she is also trying to get in to further her studies of the arcane arts, the prospect of finding a lead to their next destination as allowed them to entertain her wishes.
The guild master has of course listened to her plea, and has graciously allowed her to join the guild for an obscene amount of gold. Sadly, she, not the party, has enough gold to grant her entrance. When pressed, the guild master did mention that upon a task of great importance may consider the council to waive the fee.
It seems that their is rampant corruption occurring within the Merchant Guild, and the quarter of the city in which they pawn their wares. If the party can seek out this corruption, with proof, then perhaps this corruption, or coup can be ended quickly and those that are plotting it be brought to justice.
For this, the Wizards Guild would waive the fee for entrance.