I need to obtain sole legal custody of my daughter in order to get her passport renewed as my ex-husband lives in England and is too sick to be able to do the required documentation. He will not challenge the process but I need to know where to start.

You start with your local Family Court. Or better yet, a local Family Attorney.

Yes, you will have to file a motion in court to change the current custody arrangement ( if any) The father will be served notice and/or sent paperwork by your attorney that will have to be signed. But to be honest not sure that is not gong to be as much work for him as the passport paper work.

But after that you get a court hearing and unless the judge wants more info, they will file it. My guess to change court custody from 6 to 9 months at least.