I'm not sure if there's a way to do this, but would think there's a way to protect a PDF using your online account?
I do pay for Adobe Pro.
Is there a way to do this over Adobe's website?
Acrobat Reader for iOS or Android uses the online service to protect PDF files.
There's no way to password protect a PDF over the adobe website?
I use different laptops in the field that don't have Adobe Pro installed, and this is why I'd like to be able to login to my account, upload a file, and password the file.
Can this be done over Adobe's website?
No, this can't be done online. You need to use Acrobat to do it.
++ Adding to the correct suggestion by try67
For more information about applying password protection please look at the help article https://helpx.adobe.com/in/acrobat/using/securing-pdfs-passwords.html