Interesting idea for a fanfic, and Maggie's certainly the right character to use. Some comments:
The title is OK. Not brilliant, but not bad.
Typo: "chaffing" should probably be "chafing" ("chaff" doesn't seem to be a verb).
"Police" doesn't need to be capitalised.
"Chomping another doughnut down to let suffer, his weekly wages weren't helping matters." The first part of this sentence doesn't make a lot of sense to me. Maybe you missed out a word.
Chief Wiggum is in character.
"Dear Margaret, we need to talk." Marge suggested a few days later
That full stop should be a comma, since it's followed by a dialogue tag.
The tall figure once formerly known as Maggie, shot her mother a look
Remove that comma.
"I used to call you my sweet, you were once my sweet little angel. I raised you from infancy, and you never spoke a word against us," Marge responded, "Now look at you,
The comma after "responded" should be a full stop, since it comes before a new sentence.
"Maggie responded with a resignation" The "a" isn't needed. Also, I'd put a full stop after "resignation" since the next bit of dialogue is a new sentence.
Spelling issue: Dr. Nick's last name is spelled "Riviera", not "Rivera".
"Ah yeah, that trash-talking loon," Maggie brandished a used 547 barrel pistol
Here, on the other hand, I'd chang the comma to a full stop, since what follows isn't a dialogue tag, but a separate action.
"You always said I had a talent for... Artillery and gunfights,"
The comma at the end should obviously be a full stop. (And I'd remove the capitalisation on "artillery".) Also, there's a blank line after this line that presumably shouldn't be there.
The last bit of dialogue seems rushed and doesn't make a whole lot of sense to me (even if the lines are rather in character). I can't figure out whether Bart was on trial for thievery before or after he became a judge.
An older Homer entered through the empty corridors, "Because of the disgrace the boy had brought upon our household, out of humility he left this town!"
The comma after "corridors" should be a full stop.
Maggie laughed hollowly, "Became Chief of the Supreme Court out of pure pity,
The comma after "hollowly" should be a full stop.
Typo: "juvvie" should be "juvie"
While there isn't much so far, I think what you have is good (apart from the punctuation problems and a few typoes I've pointed out). Everyone seems in character. I especially like Marge. The dialogue is often good, if a bit verbose.
The only problem I have with it is that a few sentences, especially near the start, feel very long and a bit rambling. It's hard to pinpoint a solution: just make it a habit to read your story out loud, and if anything comes off as too shapeless, change it.
Keep going. I'm interested in seeing where this is going.