Well, it's not that a CWI CAN'T, but they are not REQUIRED to know how to write WPS's.
As to the main point of your question, ANYONE CAN write a WPS. The only requirements is that it conforms to the applicable code with all it's qualifications. They need only have the proper information to fill in all the blanks in a proper fashion that the welders can follow to accomplish the job per specs in the Contract Documents. The Fabrication Company owner, a welder, a QC dept, an outside source, etc.
The completed WPS must pass muster by the EOR for the job and then a review by the CWI on the job who is responsible to make sure all perameters, process, etc are appropriate for the materials specified for the job.
You really need B2.1, D1.1 2010, and some other resources to make sure you get all the information required.
And... WELCOME TO THE AWS WELDING FORUM!!
Have a Great Day, Brent