If you want to take legal action, hire a lawyer. If you want to puff your chest and see if they flinch here is what you do:
First let's deal with this arbitration agreement. (There is probably also a jurisdiction agreement in there, ignore that for now.) First you need to find someone to to arbitrate and a place to hold the arbitration. I suggest a babysitter, dog-walker, friend, whatever... just make sure it is a real person. You write them a letter saying that you are exercising the dispute resolution set forth in the terms and that you propose an arbitration by [insert person's name here] to take place at [location] on [date and time]. You are going to mail this to a physical address and should include a sentence asking them to let you know within 30 days if this schedule works for them. Also tell them that you prefer to communicate via email and provide an email address. Mail this letter. Just mail it plain old snail mail. In two weeks write another letter opening with a statement indicating two weeks ago you sent a letter and asked for confirmation of receipt, that you did not get receipt and would they please acknowledge. You have now puffed your chest. Pretty soon 30 days will run and you will go to the courthouse. Hopefully this will have gotten them talking to you and you can get this resolved. What I describe next is an absolute interbreeding and then slaughter of various states' court rules. I hope that more-informed stack participants edit with gusto.
If you can't resolve things go to the courthouse and tell the clerk that you want to file a claim in small claims court against [company] for [dollar amount] based on breach of contract. She will give you paperwork to fill out. You will pay her the filing fee. One of the things you will fill out is a complaint, one is a summons. It might be two in one. Here is an random example. It should be a simple form. Keep it all as simple as possible but be sure to describe the facts that support your claim. You might include the fact that they ignored your requests to arbitrate. Take this complaint and summons and serve them on your defendant. To do this you will need to get someone in the company's city to serve it. There are professionals who do this, mail or email them (if possible) the summons and complaint and pay them. They will you with provide proof of service. Now you have moved past chest-puffing and you've thrown a punch.
Now you have initiated a law suit. They must file an answer. This means that they must get lawyers involved. You've gotten their attention and it only cost you filing fees. They must now file an answer to your complaint. They will also likely file a motion to dismiss and do one of a few things: 1. complain that you need to arbitrate 2. It will complain that the contract requires legal action in a certain state or county 3. It will complain that regardless of the contract this is the wrong court for any host of reasons 4. they will claim that you haven't stated a valid complaint 5. some other boilerplate stuff. You might have a hearing scheduled to deal with these things, you might deal with them all on a trial date.
Now go back to the clerk at the court and ask for a subpoena form. You will have this served on the defendant also - it must name a person. This is a document that requires them to come to court at a time and place. Again, lawyers get involved to quash this. Lawyers = $ so you probably got their attention. Oh yeah, you need to pay a witness fee and travel costs. You can also serve a subpoena duces tecum which requires them to bring documents that you ask for. This needs to be served on the person in change of keeping records. You will probably pay the costs of the documents. The clerk will have forms for you and will describe the process in ambiguous terms while repeating "we cannot give legal advice."
You've got your documents, you have a witness coming, you are ready for trial.
If you haven't had some sort of hearing already to deal with defendant's motions to get this thing dismissed, you will need to deal with that stuff now. They will claim that arbitration is required. You tell the judge that you invited them to an arbitration that you set up and they refused to come. You show all the paperwork, including your contract with the arbiter. They will argue about jurisdiction, good luck with this one but it's a whole other problem that I can't touch here! If you survive these objections you will tell the story to the judge, you will call the witness and question her, you will present your documents as evidence. In small claims court the judge (or magistrate or referee) might ask the lawyer some questions about facts. Then the judge will bang his gavel and you wait for his decision. If you win come back here and ask how to get your money.