Can I use only OrientDB without any RDBMS?
Any DB model can store any type of data. The trade-off is what you're prepared to do in the application and what you want the DBMS to handle for you, plus performance.

So in some ways I think it would be safer to have a single system that's good enough than several disparate systems that are individually excellent at their own part but that I have to stitch together.

Think also from a business perspective. Where are the risks and opportunities here? Are you able to charge more for this new service because of the clever way you've stitched together MySQL and Orient? Conversely, if the stitching frays is there a risk of service failure and lost revenue? If you can quantify the benefits and costs, even only in tee-shirt sizes, of both approaches I think it will help you reach a conclusion.