What do we call a company that provides the services that were outsourced?

Solution 1:

In the US these (whether companies or individuals) would generally be called "contractors".

Solution 2:

The third-party companies are referred to as BPOs.

BPO is short for Business Process Outsourcer. This is a third-party service provider that handles any operations or responsibilities that a company is unable or unwilling to do in-house...

Business Process Outsourcing

the activity of paying another company to deal with tasks such as keeping accounts and paying employees:

-Cambridge online

Business process outsourcing (BPO) is the contracting of non-primary business activities and functions to a third-party provider. BPO services include payroll, human resources (HR), accounting and customer/call center relations.

Business Process Outsourcing

Another acronym in use is ITES.

These services provide a wide range of career options that include opportunities in call Centre, medical transcription, medical billing and coding, back office operations, revenue claims processing, legal databases, content development, payrolls, logistics management, GIS (Geographical Information System), HR services, web services etc.

Information Technology Enabled Services