Will log shipping work between different editions? Primary is SQL Server 2016 SE and Secondary is SQL Server 2016 EE. If not, how else can I set up high availability for this combination of servers?
There is no functional blocker in log shipping from Standard to Enterprise on the same version, however, you may want to confirm your licensing position in this scenario with your licensing reseller. Depending on how you're licensing SQL Server, you may not be covered for this additional server, even as a warm standby.
Also, you should note that Log Shipping is not a high availability technology, it is a Disaster Recovery technology. If you want high availability, you should look at implementing Availability Groups (Basic AGs are available in Standard but you can't use SE/EE together in a Basic AG architecture) or a Failover Cluster Instance.