Can we use VMDK for FCI on SQL Servers? Can I share a VMDK among 2 nodes for an active passive setup for a SQL Server Failover Cluster setup?

Most of the articles I see mention to use RDM's, however we use Veeam backup for the environment and it doesn't support RDM's.
No, you cannot reliably use VMDK files for shared storage in VMware-virtualized Windows Server Failover Clustering.  RDMs (Raw device mappings) are the only supported method of provisioning disks in this scenario. Use RDMs in physical compatibility mode (pass-through RDM). You cannot use virtual disks or RDMs in virtual compatibility mode (non-pass-through RDM) for shared storage.  You can use VMDKs for the non-shared drives.  See Setup for Failover Clustering and Microsoft Cluster Service from VMware for more details.

Since you have Veeam, you should use the SQL Server backup tools included with that package; see Veeam's help center for vSphere backup and restore for more details.