If you were an employee the Labor Code would give you relief. As an independent contractor, only the terms of the agreement between you would give you the rights you seek. You can, of course, sue your former employer and get the documents you seek through discovery in the lawsuit, but it is probably not worth that much to do so.
Good luck to you.
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If the entity you previously provided services to paid you with checks, you should be able to ask for you bank for copies of the checks, which may be a simpler first step. If you received cash, you should keep copies of all cash receipts.
In the future, you should consider keeping vendor files with copies of all payments to prevent this type of problem in the future.
Try contacting the IRS. I would assume they would be able to tell you the proper steps to take to challenge the amount listed on the 1099. This may be a better question for the tax forum. As an employee you could request the records, but as an independent contractor, I don't believe there is any requirement for them to provide copies of checks.