can someone who says they are owed money win in court without proof and what kind of proof would they need
You need just enough proof to win your case: no less, no more.
Seriously, there’s only a small subset of cases — generally those relating to real estate or long-term agreements — where a particular type of proof, like a written contract, is required. (Those cases fall under the "statute of frauds," which really is just a list of a dozen or so particular circumstances).
For every other breach of contract, like an agreement to buy equipment, or an agreement to do some minor contracting work, there’s no legal requirement the plaintiff show a writing to prove their claim.
Is a writing more persuasive than just oral testimony? Usually. But it’s not a legal requirement. The only ‘legal’ requirement in most cases is that you show, through first-hand knowledge (e.g., your testimony or someone else’s) that you’re entitled to the money you claim. Then a judge or jury will consider your testimony and the defendant’s and reach a factual conclusion.