sudo packer build docker.json

docker.json

{
  "variables": {
    "ansible_host": "default",
    "ansible_connection": "docker"
  },
  "builders": [
    {
      "type": "docker",
      "image": "centos:7",
      "commit": true,
      "run_command": [
        "-d",
        "-i",
        "-t",
        "--name",
        "{{user `ansible_host`}}",
        "{{.Image}}",
        "/bin/bash"
      ]
    }
  ],
  "provisioners": [
    {
      "type": "ansible",
      "groups": ["webserver"],
      "playbook_file": "./webserver.yml",
      "extra_arguments": [
        "--extra-vars",
        "ansible_host={{user `ansible_host`}} ansible_connection={{user `ansible_connection`}}"
      ]
    }
  ]
}

webserver.yml

- name: configure webserver
  hosts: webserver
  tasks:
    - name: install Apache
      yum:
        name: httpd

packer build docker.json for the following content works without sudo.

{
  "variables": {
    "ansible_user": "root",
    "ansible_host": "default",
    "ansible_connection": "docker"
  },
  "builders": [
    {
      "type": "docker",
      "image": "centos:7",
      "commit": true,
      "run_command": [
        "-d",
        "-i",
        "-t",
        "--name",
        "{{user `ansible_host`}}",
        "{{.Image}}",
        "/bin/bash"
      ]
    }
  ],
  "provisioners": [
    {
      "type": "ansible",
      "groups": ["webserver"],
      "playbook_file": "./webserver.yml",
      "extra_arguments": [
        "--extra-vars",
        "ansible_user={{user `ansible_user`}} ansible_host={{user `ansible_host`}} ansible_connection={{user `ansible_connection`}}"
      ]
    }
  ]
}

This also works.

{
  "variables": {
    "ansible_host": "default",
    "ansible_connection": "docker"
  },
  "builders": [
    {
      "type": "docker",
      "image": "centos:7",
      "commit": true,
      "run_command": [
        "-d",
        "-i",
        "-t",
        "--name",
        "{{user `ansible_host`}}",
        "{{.Image}}",
        "/bin/bash"
      ]
    }
  ],
  "provisioners": [
    {
      "type": "ansible",
      "user": "root",
      "groups": ["webserver"],
      "playbook_file": "./webserver.yml",
      "extra_arguments": [
        "--extra-vars",
        "ansible_host={{user `ansible_host`}} ansible_connection={{user `ansible_connection`}}"
      ]
    }
  ]
}