They can only if there are rooms available. Disney sets a certain amount of rooms aside for the free dining promotions. If the free dining is released, you have to cancel your existing reservation and book one of the free dining rooms. Even if you already have an existing reservation, it doesn't matter as you have to book a new room anyway. That being said, we usually go ahead and book a room anyway. We go the same time every year, and will go with our without free dining. So, we book a room at the resort we really want to stay at just in case free dining doesn't fall in to our travel dates. We have been able to get free dining every year though. Two years ago we had originally booked a standard room at port orleans riverside. However, when free dining came out, the only rooms available were the royal rooms. We didn't really want to pay that much for the room, but decided to go ahead and do it since we really wanted to try the resort. We are glad we did, but I could have used that extra money on souvenirs

Not only does Disney only release a set amount of rooms at each resort for free dining, they may only release specific booking categories at each resort (some may only have preferred rooms available for free dining, or only water views, etc). They use free dining to fill rooms that aren't booking up to their expectations. So, in theory by booking early it could hurt the chances that free dining is even offered for a specific resort or booking category. I hope that makes sense. In short, yes, your TA can rebook you under free dining if rooms are available