取引先って英語でなんて言うの？

A 'business associate' is someone that you do business with and can be in any sector.
A 'client' is a person or business that you are providing a service for. For example a lawyer will have many different 'clients'.
A 'business partner' is usually someone that you run your company with. A partner in a company will usually have a similar amount of pull in the company as yourself. This however can also be used to describe someone from another company that you are working alongside with.

A company may do business with another company in two ways. The company may be a supplier of goods or services to the other company. In this case, the company is a supplier/service provider and the company being supplied with goods or services is a customer or a client. If the company is the receiver of goods or services, it then becomes the customer or the client.
So, you may say:
We do business with a lot of customers/clients. (If you are a supplier)
or
We do business with several suppliers/service providers. (If you are a customer/client)

In a sales organisation, someone to whom you sell goods or services would be known as either a customer or a client.
A person or company that sells goods to you is known as your supplier.
"We will receive a delivery from our supplier later today."

The term most commonly used for the business/company you deal with is called a client you could also called the person in the company a business associate you could also be called a customer if you are buying goods/services from the business
The term business partner can also be used to say a person that works with you running your company as well

There are a number of different ways to refer to a business that you have a working relationship with. The correct words depends on what kind of business relationship the two companies have.

For example, if the company provides a service to the other, it is most common to refer to them as a 'client' or a 'customer'. An example of this would be how an outsourced cleaning company would refer to a business's office that they are responsible for cleaning.

However, if you are a business that receives a service from another, you would refer to them as a 'supplier'. Using the same example as above, the business that operates in the office and uses the cleaning service would call them a 'supplier' as they supply the service to that business.

To describe a company that your company does business with might be called a ‘business associate’ or ‘business partner’, especially when the business is being done on equal terms. However, if one company is being employed by another, e.g. a law firm is being employed by a retail company, we might say the retail company is the client of the law firm, meaning that the retail company is paying the law firm to do a job.