How to design a database with one-to-many and then many-to-many relationship?
This is a many to many relationship. You will have a Customer table and a TruckDriver table.

You will also have a table called Delivery or similar which will manage the relationship between the two.

The delivery table would have the columns

CustomerID (foreign key, referencing the PK in the customer table)
DriverID (foreign key, referencing the PK in the driver table)
Rate


If one customer can recieve an offer from the same driver more than once, it may also be worth adding a column DeliveryID as the Primary Key on the table, otherwise you can use a composite key of CustomerID, DriverID

There may be other information you want to include in your delivery table such as date of offer and a flag as to whether the offer was accepted or not but that is not specified in the question.