Legal Question in Business Law in California

Whether you have any meaningful recourse depends on factors not covered in your question. Absent a confidentiality agreement, your account history with a vendor is not private or privileged information. It is unclear whether "shared some assumptions" were matters of opinion or actual false assertions of fact. Only false factual statements are actionable. Then there is the question of what results this had. Unless it actually damaged your relationship with the customer in a way that can be proved in court, you have no recourse because you have no damages. Defamation is a "no harm, no foul" area of the law. So, if there were false statements of fact from the vendor to the customer AND it caused you to lose the customer or other demonstrable actual losses or harm to your business, then you have recourse in the form of a lawsuit for damages for defamation.

The other recourse you might have is if you believe and can demonstrate that harm has occurred and will continue, or has not yet occurred but most likely will, from the vendor spreading false information about your company, then you can sue for an injunction against that conduct.