I am preparing my Apple Watch for trade-in. Do I need to send the charging cable with it?

That is ultimately up to you to decide:

In some cases, your final trade-in value can vary depending on if you send in your power adapter or not. You don’t have to, but you can always send in your chargers and cables, and we will recycle them for you.

Apple Trade In - Apple

No, do not send any bands or accessories, only send the Watch itself.

The reference to the trade-in value is primarily for Macs.