What you are really asking is "How can I see what people are replying to on their screen before they send"
The problem is, GMail does not have some kind of live-updating way of sharing an individual account status with other accounts
BUT there are other Google Products that do. For example:
- One solution is you could have a separate Google Spreadsheet where people copy and paste the subject line of the email they are replying to and put it in one cell, the customer's name/email in the next cell, and finally their name in the cell next to it. Google Spreadsheets live updates, and you can see who has grabbed what email. Every day or every morning you can delete everything in the spreadsheet and start over
- You can do the same with Google Docs. You could even have people draft their emails in a single shared Google Doc (assuming there are not TONS of emails coming through at once) and this way if someone is about to reply to an email and you want to add something, you could collaboratively edit it as well.
A true ticketing system is probably better, but these are quick ways you could deal with it. I personally sometimes will reply back to my team (but not the customer) with a quick "I'll get this one" email - that way everyone knows I have it and they don't have to worry about it