Do I need to install a license key in vCenter to get NAT and firewall working?
If so, where do I go to get this key?
vCenter does not provide NAT functionality - as for firewall functinoality by itself it does not provide any protection but you can implement vShield which provides endpoint/firewall protection - vShield is included with all vSphere license levels.
vCloud requires vCloud Network & Security (vCNS) to be setup. This requires a vCNS or vCloud Suite licences be setup in vCenter. (In each vcenter that will be in your cloud).
Normally when getting a vCloud eval you will be given a key that will allow for this - but this key does need to be added to vCenter (the base vCloud install does not add it for you).