Because during the summer we have so little time, you will develop a relatively light, non-standard requirements document consisting of just use cases and associated wireframes.
A narrative description of the purpose of your web site. Include a description of your web site's audience(s).
Develop a set of use cases for each type of user that your site supports. For a good description of how to write a use case, see https://www.usability.gov/how-to-and-tools/methods/use-cases.html. You will create "heavyweight" use cases.
Example types of users: shoppers, store managers, game players, etc.
Example use cases:
For each of your use cases create a UML use-case diagram. This will help you identify subcomponents of your system. If you are unfamiliar with UML use-case diagrams, there is a good explanation at http://www.uml-diagrams.org/use-case-diagrams.html. One of the examples there is for online shopping.
Next, for each of your use cases develop a wireframe of the web page for each major step in the use case. ("Major" in the sense that the page's appearnce changes significantly.) The wireframes can be hand-drawn, or use Powerpoint, or a graphing program. There's a nice overview of wireframes at https://www.infoq.com/articles/wireframes-start-development-projects.
Provide a timeline for the remaining weeks of the semester. The timeline should include at least two "deadlines" for each week. Clearly specify which of your teammates is responsible for completing which work by each deadline. Use a Gannt chart to clarify the responsbilities and deadlines.
Integrate PayPal into your system. The developer.paypal.com site provides lots of documentation for how to do this, including integration of their own shopping cart technology (if you want to use it) and use of the PayPal Sandbox.