Can I charge shipping based on the weight of the order?

Yes, with a Weight-based (brackets) method in the Shipping Zone plugin.

1. Open the Shipping Zones page and edit (or create) a zone, then add its Regions.
2. Under Shipping methods, click + Add method and set Type to Weight-based (brackets).
3. Add brackets with + Add bracket: a Min kg, a Max kg (blank means no upper limit) and a Cost, or tick Free for that band.
4. Click Save zone.

The cart's total weight is matched against your brackets, and the first bracket that fits sets the price.

One caution: the current storefront cart doesn't pass each product's weight into this calculation, so an order can be priced as 0 kg (your first bracket). Place a few test orders and check the shipping before relying on weight brackets. Price-based or Quantity-based brackets don't depend on weight.