How to pass array to Ansible URI module
Your letters variable is a list, but when you write:
"{{ letters }}"

You are asking -- explicitly -- to render it as a simple string. You don't want that; you want to maintain the structure of the data. You can do that like this:
- name: Create a letter job
  uri:
    url: 'http://localhost:8000/api/app'
    method: POST
    headers:
      Content-Type: application/json
    body: '{{ { "letters": letters } }}'
    body_format: json
    return_content: yes
  register: response

Here, we're using a Jinja template ({{ ... }}) to create dictionary with a single key, letters, whose value is the content of your letters variable.
Using this sample application:
from pydantic import BaseModel
from fastapi import FastAPI


class Letters(BaseModel):
    letters: list[str]


app = FastAPI()


@app.post("/api/app")
def echo(letters: Letters) -> Letters:
    return letters

And the following playbook:
- hosts: localhost
  gather_facts: false
  tasks:
    - name: Create a letter job
      uri:
        url: 'http://localhost:8000/api/app'
        method: POST
        headers:
          Content-Type: application/json
        body: '{{ { "letters": letters } }}'
        body_format: json
        return_content: yes
      register: response

    - debug:
        var: response

We see as output:
TASK [debug] ********************************************************************************************
ok: [localhost] => {
    "response": {
        "changed": false,
        "connection": "close",
        "content": "{\"letters\":[\"aa\",\"ab\",\"ac\",\"ad\",\"ae\"]}",
        "content_length": "38",
        "content_type": "application/json",
        "cookies": {},
        "cookies_string": "",
        "date": "Fri, 02 Jun 2023 16:42:50 GMT",
        "elapsed": 0,
        "failed": false,
        "json": {
            "letters": [
                "aa",
                "ab",
                "ac",
                "ad",
                "ae"
            ]
        },
        "msg": "OK (38 bytes)",
        "redirected": false,
        "server": "uvicorn",
        "status": 200,
        "url": "http://localhost:8000/api/app"
    }
}

...which I think is exactly what you were after.

I'm trying to pass an array using extravars to the Ansible URI module.
% ansible-playbook someplaybook.yml -e '{ "letters": [ "aa", "ab", "ac", "ad", "ae" ] }'

When I run it through the URI module, I'm getting [ "['aa', 'ab', 'ac', 'ad', 'ae']" ] in the response. How do I send the entire array without the brackets and quotes. I've tried "{{ letters | replace("[", "") | replace("]", "") }}" but that doesn't get rid of the exterior quotes.
  - name: Create a letter job
    uri:
      url: 'http://localhost:8000/api/app'
      method: POST
      headers:
        Content-Type: application/json
      body: '{ "letters": [ "{{ letters }}" ] }'
      body_format: json
      return_content: yes
    register: response

Current Output:
"response": {
    "id": 2796831356421,
    "letters": [
      "['aa', 'ab', 'ac', 'ad', 'ae']"
    ],
    "lastError": null
}

Desired Output:
"response": {
    "id": 2796831356421,
    "letters": [
      'aa', 'ab', 'ac', 'ad', 'ae'
    ],
    "lastError": null
}

