Is it permissible to copy customers' numbers—including those who come to the shop or those who deal online—to an employee's private phone, knowing that the employer has not explicitly forbidden it, and that the purpose of doing so is to build relationships to ensure professional continuity?

There is no harm in keeping the phone numbers of clients and customers on your personal phone, unless the employer explicitly or implicitly (by custom) stipulates that customer data should not be used. In that case, the condition must be adhered to, because the Prophet, peace and blessings be upon him, said: “Muslims are bound by their conditions, except for a condition that prohibits what is lawful or permits what is unlawful.” This is also based on the rule: "What is known by custom is like what is stipulated as a condition."