No, you will continue to have service with your current carrier up to the moment you're live on Reach. We do ask that you remain on Wi-Fi during activation so you receive notifications alerting you when the port is complete and insert your Reach SIM.
In case you're interested in the details:
When you input your current carrier's account, PIN and billing zip code into the app, we send a request to your carrier asking them to release your number to us. This is the part that can take 5 minutes to 2 hours (and in some exceptional cases we have seen this take a few days with landlines, but this is not common).
While we are waiting for your carrier to release your number, your old service continues to work with no interruption. It will continue to work until the moment you receive an in-app notification and email that your Reach activation was successful. Once you're alerted that your activation was successful, your old carrier SIM will no longer work.