TypeError: Cannot read property '$emit' of undefined

In the script

Very strange, can you show the relevant code?

User.register(newUser).then((data) => {
        if (data.data.user) {
          this.$emit('event');
          toast.add({
            severity: "success",
            summary: "Success Message",
            detail: "User Registered",
            life: 3000,
          });
        } else {
          toast.add({
            severity: "error",
            summary: "Error Message",
            detail: data.data[0],
            life: 3000,
          });
        }
      });

I just want to close the parent sidebar after registering.

You need to share the wider context of the Vue component so we can understand what this should be.

<script>
import { Form, Field } from "vee-validate";
import * as Yup from "yup";
import User from "../../../apis/User";
import { useToast } from "primevue/usetoast";

export default {
  components: {
    Form,
    Field,
  },

  setup() {
    const toast = useToast();

    const schema = Yup.object().shape({
      name: Yup.string().required("Name is required"),
      email: Yup.string()
        .required("Email is required")
        .email("Email is invalid"),
      password: Yup.string()
        .min(8, "Password must be at least 8 characters")
        .required("Password is required"),
      password_confirmation: Yup.string()
        .oneOf([Yup.ref("password"), null], "Passwords must match")
        .required("Confirm Password is required"),
    });

    const onSubmit = (values) => {
      // display form values on success
      console.log(values);
      let newUser = {
        name: values.name,
        email: values.email,
        password: values.password,
        password_confirmation: values.password_confirmation,
      };

      User.register(newUser).then((data) => {
        console.log(data);
        if (data.data.user) {
          this.$emit('event');
          toast.add({
            severity: "success",
            summary: "Success Message",
            detail: "User Registered",
            life: 3000,
          });
        } else {
          toast.add({
            severity: "error",
            summary: "Error Message",
            detail: data.data[0],
            life: 3000,
          });
        }
      });
    };

    return {
      schema,
      onSubmit,
    };
  },
};
</script>

Seems you need to pass the context into the setup function:

https://v3.vuejs.org/guide/composition-api-setup.html#context

setup(props, context) {
	// ...

      User.register(newUser).then((data) => {
        console.log(data);
        if (data.data.user) {
			context.emit('event');

			// ...