There are non-profit legal organizations and legal aid groups, but they are few and far between and are very selective in the types of cases they choose to represent. I don't think you'll have much luck finding it.

The fee the lawyer can charge is set by SSA law. The fee agreement has to be approved and the lawyer cannot legally charge more than what SSA approves. They are allowed to charge you for money they expend on yoru behalf (photocopies, medical exams) but not for time or expertise. The amount in most fee agreements is 25% of retroactive benefits, not to exceed $6000 (currently).

Lawyers are in business so they do want to make money and some will not want to be bothered taking on a case at the initial level because the fee will not be high if the case is approved in a two or three months. There are a few that might be willing to give your case a little bit of attention at the initial level, but lawyers like to make deals with other lawyers which is really what is done when a denied case goes before an Administrative Law Judge. The process takes a long time, so the retroactive benefits keep adding up and the fee may hit the max $6000. Or, the ALJ may amend the date of onset; there may be significant Worker's Comp offset, or a number of other issues which would reduce the retroactive benefits and the attorney fee. But a rep could also put in two or three years of waiting for a decision and put in hours and hours of work by him and his staff, and end up with zero fee, work for nothing.

First thing to do is make sure you fully complete the disability report and don't assume that SSA will read between the lines or that SSA will beat the bushes looking for your medical records. Start a claim with being thorough and don't expect the records will speak for themselves. Just because you told your doctor something doesn't mean that it was wiritten down.

Hire a rep now, later, or never. Your decision. All are valid.