A W-9 is given to Independent Contractors, which is anyone who performs work without being hired as an employee. People who get jobs through a staffing agency are employees of the staffing agency until some contractual period of time has passed, and as such would be given a W-4 from the staffing agency.
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Texas has different classifications of non-employees depending upon the type of company you work for. If you work for a staffing company, you are referred to as a "temp" or "temporary employee" and are assigned to the client company for a short period of time. The staffing company is your employer.
If you are assigned to a client company for a long period of time, you may be a "leased employee" and the company you work for may be a Texas-licensed staff leasing company. There are also co-employment and joint employment arrangements in which the client company and staffing company mutually decide which party will be responsible for certain employment functions. The Texas Workforce Commission website has legal descriptions of the different types of non-employee arrangements in addition to a description of independent contractors.
In Texas it is very difficult to classify anyone as an independent contractor if they are told when to work, where to work, and provided what they need to do the work. Even if they only do the work for a short period of time, they would be an employee.
DISCLAIMER: This is not specific legal advice and does not establish an attorney/client relationship.