What are the purposes of Reply messages in Lamport Algorithm for Mutual Exclusion?
I believe the purpose is to fulfill the first condition of entering the critical section (site $S_i$ must receive a message from all other sites with timestamp greater than its own request) in case some other site never requests to enter the critical section. 

If $S_i$ sends a critical section request to $S_j$, but $S_j$ never requests the critical section and doesn't send a reply, $S_i$ will never receive a message from $S_j$ thus causing it to wait forever to enter the critical section. In this scenario, $S_i$ cannot know for sure that $S_j$ does not have a pending request with a timestamp lower than $S_i$'s request.