You need an attorney who is licensed to practice law where you live.
Also most attorneys will speak with you for about 10 to 20 minutes at no charge to assess the merits of your case.
I recommend that you go to the Yellow Pages and look up the section for attorneys.
since this is a personal/family issue I recommend that you call several attorneys who practice Family Law.
From the website, it looks like they are a PI firm snooping on juries and judges. Not sure there would be a lot of demand and it seems borderline unethical, unless an attorney has a legit reason to do a background check on a juror to disqualify them. Usually a lawyer doesn't have time to do that while a trial is going on. Judges the attorneys usually already know about, from experience or talking to other judges. So I don't know how much demand there would be except for really big long term trials, which are rare.
Prior verdicts can be found through legit subscription websites. There are agencies that do sample trials so attorneys can see how the concept of their case, their client, their opening and closing statements go over.
I'm a retired lawyer now working as a paralegal. For whatever reason, I get the cold calls in our firm. I jot down the information and pass it on to the attorney who buys services for the firm. They generally get ignored at that point.
Most of those services are sold by word of mouth. Attorneys ask each other and ask questions on listserves. So I doubt that you would get very far cold calling, unless you can name references from other attorneys in the same field.
Go to the interview and see what that's about. They may do it on a Sunday because they are very busy, which is why they are hiring, or because most of the people they interview are working and they can come on a Sunday.