Nope, you do not need to have Google Classroom or any other LMS to use Classwork.com. 

When you create an assignment in Classwork, you'll be provided with a link to send to your students. All your students need to access the assignment is this link. How you send it to them does not matter. You can copy the link and send it to your students via email, text message, displaying it on a whiteboard, etc.

To access and complete the assignment, your students will simply click on the link. There are settings in the Classwork Assignment screen that let you collect their name, ID, or email so you can keep up with your classes without an LMS or Google Classroom.

Classwork.com and Google Classroom play nicely together, however, and so you may want to check it out... just sayin'!