I have a simple question. Does a vetiver model object carry individual patient data from the dataset it was fitted to? Would really appreciate any help on this.
It really depends on what the underlying model you are using does!
The vetiver package itself removes model components that are not needed for prediction via butcher. You can see, for example, what is removed from an lm() model or a ranger random forest model. Some types of models do still include the training data after butchering. As an example, lm() and friends carry around their training data because it is needed to generated prediction intervals; see related discussion here.
The vetiver package does not take any steps to additionally keep training data in the model object, beyond what the underlying model already does.