An essential component of your contract with them is that they will provide you with food free of animal stuff (the exact nature of "vegan approved" may be up for debate, but actual meat should not be included). So they breached their contract with you, and you might sue them for breach of contract.
The case of Gupta v. Asha (orders were mixed up) could be useful in this matter. In this case, Hindus were served samosas containing beef, despite repeated assertions that they were vegetarian. Plaintiffs sued, the case was dismissed, and the appeals court deemed that the lower court was part right and part wrong.
The important thing to understand is that you have to claim a specific legal wrong (and your attorney would advise you about that). Those plaintiffs made a claim under NJ product liability law, and that was dismissed because the product wasn't defective, it was simply the wrong product.
They also made a claim regarding fraud, and there was at least a legal question behind that claim, but because there was no ascertainable loss (e.g. the plaintiffs were not hospitalized because of the omission), so that claim too was dismissed. Likewise a negligence claim was dismissed.
Plaintiffs did get to first base w.r.t. the claim of emotional distress arising from breach of contract. I have no evidence regarding the final disposition of the case: the point is that there might be a legal basis for a lawsuit in your state, but you would need to secure the paid assistance of an attorney.