Deliveries proceed quickly in storm (7)
I believe the answer is:
overrun
'storm' is the definition.
(thesaurus)
'deliveries proceed quickly' is the wordplay.
'deliveries' becomes 'over' (I've seen this before).
'proceed quickly' becomes 'run' (I've seen this before).
'over'+'run'='OVERRUN'
'in' is the link.
(Other definitions for overrun that I've seen before include "Cover - attack" , "Go on too long (of a performance)" , "Exceed (eg, time, cost)" , "Plagued by" , "Heavily defeat" .)