I disabled MMS in Messaging on my phone. Is FB Messenger vulnerable? If so, what can I do to minimize the risk? (I couldn't find anything about MMS in FB Messenger)
libStageFright is a general media library used widely on Android. Facebook Messenger is likely to interact with it for media, and thus you are vulnerable through it.
Your best bet is to update — to a custom ROM if the manufacturer/carrier hasn't made a patch available. Otherwise, see Stagefright security issue: what can a regular user do to mitigate the issue without a patch?