What open-source codebase/software would allow my customers to create their own landing page? My goal is for them to answer questions and select content that would auto-populate to their page, but also have the ability to add their own content.
You could use something like a Website as a Service (WaaS) that allows you to create a platform that users can sign up to and pay you monthly for a service. You just need a domain name (example.com) and then users would get a subdomain (user.example.com) or they can register their own domain which you can also make profit on.
I would imagine a simple sign up process and a dashboard that allows your users to either use basic drag and drop tools or simply a form if you're going to do that for them. It basically sounds like a mini-website or a one-page site which is perfectly doable with WaaS using WordPress on a VPS or Cloud Computing, which can help with expansion when needed. There are other alternatives as well but this one is open source and allows you to easily accept payments for their service.
You can eventually add features to the service like private content and even allow your customers to add external content like Instagram/ Twitter. There are thousands of possibilities.
Do let me know if you need further help or advice, or help setting this up, as it's something I have experience with.
I know these aren't open source, but don't forget that Startups.com has https://www.launchrock.com/ which is basically what you are offering and you could create a form then create it for them, that way you are providing a design service that you can choose what to charge, rather than providing the entire platform. https://simvoly.com/whitelabel-website-builder is a white label drag and drop builder that you can launch as a service to your customers.