Medical emergency caused by a speaker's inactivity (6)
I believe the answer is:
arrest
'medical emergency' is the definition.
(as in cardiac arrest)
'a speaker's inactivity' is the wordplay.
'a speaker' becomes 'ar' (I am not sure about this - if you are sure you should believe this answer much more).
'inactivity' becomes 'rest' (rest is a kind of inactivity).
'ar'+'rest'='ARREST'
'caused by' acts as a link.
(Other definitions for arrest that I've seen before include "Catch" , "apprehend a law-breaker" , "Collar" , "Land next to the sea" , "Stop (on suspicion)" .)