Actually, Mr Packethead is 'mostly' correct - PCBway actually do RigidFlex, BUT their Online Ordering system is only for Flex PCBs.
You will need to upload your file via that GUI and place implicit instructions for them to contact you regarding details PRIOR to commencing production. They will then send it out and return a quote for you. Their Flex PCB cost is very reasonable.
Re: FFC connectors - we currently use connectors [28way, low profile, 0.5mm pitch], but require the board space for other things now, hence the move to RigidFlex. We will still need to use a connector on the 'other' side however.
I'm not certain who they use for their RigidFlex manufacturing, but we are about to shortly discover what their product is like - in the next week or so.
There is a stack of info online, and most of it is both contemporary and consistent. Wurth also produce a RigidFlex Guidebook which is great value.
Good luck - Peter
pcbway.com do them, quote it online. Will give you an idea.