Do I need to remove the Verizon SIM card from my iPhone before returning for trade-in?

YES, if you want to use the Verizon SIM card for your new iPhone, I mean keep the same phone number.

That's easy, then you don't have to edit your Apple ID details. -> ‘trusted phone numbers’.

If the new iPhone use a different SIM card format, you can ask Verizon for the correct format. Normally it is free.