When it comes to building an online shop for a business that deals with heavy materials for construction, there are a few website builders that can meet your requirements. Here are two options worth considering:

1. WooCommerce:
WooCommerce is a popular e-commerce plugin for WordPress websites. It offers a range of features and customization options that can cater to the specific needs of your heavy materials business. You can easily manage product variations, set weight-based shipping rates, and integrate with various payment gateways. WooCommerce also provides a wide range of themes and extensions to enhance the functionality of your online shop.

Example: Check out the website of "The Steel Store" (https://www.thesteelstore.com/), which uses WooCommerce to sell heavy steel materials. They have customized their shop to include weight-based pricing and shipping options.

References:
- WooCommerce website: https://woocommerce.com/
- WooCommerce documentation: https://docs.woocommerce.com/

2. BigCommerce:
BigCommerce is a robust e-commerce platform that offers advanced customization options and scalability. It provides features specifically designed for businesses dealing with heavy materials and complex order management. You can easily set up weight-based pricing, manage inventory, and integrate with shipping carriers to handle deliveries. BigCommerce also offers a wide range of responsive themes and has a user-friendly interface.

Example: Take a look at "Construction Materials Direct" (https://www.constructionmaterialsdirect.com/), which uses BigCommerce to sell heavy construction materials. They have customized their shop to include weight-based pricing and a streamlined checkout process.

References:
- BigCommerce website: https://www.bigcommerce.com/
- BigCommerce documentation: https://support.bigcommerce.com/

Both WooCommerce and BigCommerce provide extensive documentation and support to help you get started and customize your online shop according to your specific needs. It's important to evaluate your requirements and compare the features and pricing of these platforms to determine which one suits your business best.