The best post I've seen on this is a fix that would change the client to post all chat to the client from the servers. It currently posts chat from your local computer directly to your client chat log, but everyone else sees it after it pings the server. So two people can call it and actually see different orders bc of this issue, but the other 3 players will see the same order of whoever posted to the server first. By changing to make the local text come off the server instead of direct posting to the chat log, suddenly the 5th player would see things in the same order as everyone else.
If you just delay people being able to post, then with the current system then everyone would post as soon as it opens (since everyone would have time to type/paste their call in and wait to be able to post just spamming enter the whole time, and as a result, everyone would see their own call first (since it doesn't post to server and immediately shows up in chat history) and everyone else behind them (since they see other people's calls after they come from the server). Delaying when people can make calls w/ the current system would actually make it so that everyone almost always thought they called it first and the other person didn't bc that's how chat logs would read.
Also it would probably just lead to people locking stuff instead of calling it if you were able to lock before chatting, in which case why not just have draft w/ only one side, and no bans? (IE, fp gets to lock, then second gets to lock, etc. Then instead of call order it becomes pick order, but people can still get to game faster than draft bc there is no alternating or bans, just one lock then the next until the team is filled)