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
As the previous answers and comments point out:
Pick any two; with all three it's a no-go. @Jimbo's answer is correct (and the proper way to do it), but excludes #3. Throwing out #1 means going back in time. But, we can throw out #2, and go directly to the source:
curl -s https://twitter.com/EA_FIFA_FRANCE |
sed -ne 's/^.*ProfileAvatar-image.*\(https:[^"]*\).*$/\1/p'
The sed command just says, find the line that contains "ProfileAvatar-image" and print the substring that looks like a quoted URL.
This is less stable than an authenticated API call, since Twitter may change their HTML at any time, but it's easier than dealing with OAuth, and no official rate limits!
The PHP translation should be straightforward.