In API 1.0, we can use users/profile_image/:screen_name
For example : http://api.twitter.com/1/users/profile_image/EA_FIFA_FRANCE
B
You say you want to use Twitter API 1.1 and yet you don't want to authenticate your requests. Unauthenticated requests are not supported in API v1.1. So please adjust to the API change. See updates :
You can get image from profile_image_url field of https://api.twitter.com/1.1/users/show.json request. Either a id or screen_name is required for this method. For example :
GET https://api.twitter.com/1.1/users/show.json?screen_name=rsarver
See details here https://dev.twitter.com/docs/api/1.1/get/users/show