Webcam for Proctoring

[AwardTour]

...  I ended up getting a new laptop with a built-in camera....

and they let you use that for proctoring ?  

I thought the camera had to be free standing so you could pick it up and move it around to show the surroundings

WGU now requires external webcams                  
https://cm.wgu.edu/t5/Academic-Requireme...-714347137

External Webcam Minimum Specifications
Students beginning their programs on October 1, 2019 or later are required to obtain an external webcam that meets the following technical specifications. If you received a webcam from WGU, prior to October 1, 2019, you will still be able to use that webcam, or a different external webcam that meets the specifications below.

Note: A laptop's integrated camera does not meet these requirements. Webcams must be external and elevated to achieve required viewing angle.

Webcam Tripod 
To ensure the integrity of the assessment experience, webcams must be elevated 8 inches above the desktop to allow online proctors a view of a student’s work space and surrounding environment. If the webcam selected doesn’t include an elevated option, a tripod is required for use with the webcam to meet technical requirements for proctored assessments.

this is one of WGU's recommended webcams            
https://www.amazon.com/Logitech-Laptop-W...way&sr=8-1

and these are two of the recommended tripods                  
https://www.amazon.com/Lightweight-Webca...pons&psc=1

https://www.amazon.com/AmazonBasics-WT03...pons&psc=1

I was just telling the OP to avoid the Logitec C270's because the OP mentioned it. I am taking exams at Study.com so a built-in cam is good enough for me right now. Your list of recommended WGU cams should do the job.