Yes and no.
Depending on your plan you can have edge under contract or off contract.
If you are at a certain point in your contract and you can trade in your phone for an edge agreement which will drop your contract. If you are not at that point you can edge , but you will not get the edge or off contract discounts.
If you log into My Verizon it will tell you your contract end date. If you get on Early Edge yes that ends your contract. That is why you have to turn in your current device. Your doing that instead of paying a ETF. Typically you're only eligible for Edge if your off contract. If you have at least a 6 GB plan and you can't pay full price all at once for a device Edge is better than a 2 year contract