It must be possible: there are businesses built around selling open source systems.

However, I personally don't see it as a particularly attractive business model. On the one hand, by "crowd-sourcing", I imagine you mean "farming out development work to unpaid volunteers". On the other hand, you'd need to have some kind of built-in reason why paying customers should pay you, instead of just using the (freely available) open source themselves.

Let's turn the question around: why would you want to try to "build a business to produce and sell software or a subscription web app" that is open source?