How to add items to the cart, this example is headless but can be modified for a regular setup by also posting a CSRF token.

Craft Add Items To Cart

import axios from "axios";

async function request(request) {
  if (!request) {
    return { data: undefined, error: "no request" };
  }

  const data = typeof FormData !== "undefined" ? new FormData() : {};
  if (request.data) {
    Object.entries(request.data).forEach(([key, value]) => {
      data.append(key, value);
    });
  }

  return await axios({
    url: request.url,
    method: request.method || "post",
    data,
    headers: {
      Accept: "application/json"
    }
  })
    .then(d => {
      return { data: d.data, error: undefined };
    })
    .catch(error => {
      return { data: undefined, error };
    });
}

const saveNewLineItems = async (cartNumber, newItems) => {
    const method = "POST";
    const payload = {
      action: `commerce/cart/update-cart`
    };

    if (cartNumber !== undefined) {
      payload.orderNumber = cartNumber;
    }
    
    // payload[window.Craft.csrfTokenName] = window.Craft.csrfTokenValue; // Append CSRF Token if not headless

    if (Object.keys(newItems).length) {
      Object.keys(newItems).map(function(key, index) {
        const item = newItems[key];
        payload[`purchasables[${index}][id]`] = item.purchasableId;
        payload[`purchasables[${index}][qty]`] = item.quantity;
        payload[`purchasables[${index}][options][Size]`] = item.size; // optional
      });
    }

    const { data, error } = await request({
      url: `${process.env.api}`,
      method,
      data: payload
    });

    if (!data || error || !data.cart) {
      return Promise.reject();
    }
    if (data.success) {
      return data.cart;
    }
  };