Can an ECS Without an EIP Bound Access the Internet?

Yes.

You can use the NAT Gateway service to allow ECSs in a VPC to access the Internet using an EIP. The SNAT function provided by the NAT Gateway service allows the ECSs in a VPC to access the Internet without requiring an EIP. Additionally, SNAT supports a large number of concurrent connections for applications that have a large number of requests and connections. For more information about NAT Gateway, see NAT Gateway Service Overview.