From B2BCommerce (Commerce on Core) for Salesforce, using the LWC (Lightning Web Component) within the LWR, this gist will call a commerce-api URL with the session credentials of the running user.

fetchCommerceUrl.js

/* Returns a promise */
fetchCommerceUrl(url) {
    let apiver = 'v64.0';
    let commercebase = '/webruntime/api/services/data/' + apiver + '/commerce';
    let fullurl = commercebase + url;

    console.log('Begin fetchCommerceUrl on ', fullurl);

    return fetch(fullurl, {
      method: 'GET',
       headers: {
         'Content-Type': 'application/json'
       },
       credentials: 'same-origin'
    })
    .then(response => {
      return response.json();
    })
    .then(data => {
      if (data) {
        return data;
      }
    })
    .catch(error => {
      console.log('Error with fetchCommerceUrl: ', fullurl, error);
    });
  }

Example usage:

async fetchLegacyOrders() {
    let legacyStoreId = '0ZEbb00000014PlGAI'
    let legacyOrderSummaryUrl = '/webstores/'
    + legacyStoreId + '/order-summaries'
    + '?fields=GrandTotalAmount%2CDeliveryBlock__c'
    + '&pageSize=100'
    + '&ownerScoped=false'
    + '&language=en-US'
    + '&asGuest=false'
    + '&htmlEncode=false';

    const legacyOrders = await this.fetchCommerceUrl(legacyOrderSummaryUrl);
    console.log("%c legacyOrders", "color: green", legacyOrders);
  }