Is it possible to specify which network or NIC to use when running a backup?

Are they on different sub-nets? Are the computers that you want to back up on these separate networks ... or are the storage servers on these separate networks?

The Acronis server which also has the storage node which contains the vault, and the server(s) I want to backup have NICs connected to both the production and backup subnets.

So in other words ... you are running 2 NICS on each machine for redundancy? It would be great to see a diagram.

There's no redundancy. 2 separate subnets. The second subnet is used for backup traffic. No diagram necessary.

Is it possible to direct Acronis traffic to a specific subnet?

That's clearer. By my understanding of your description the only way to do this is to put the storage node(s) only on your backup subnet.
This would require a separate machine. As long as the storage node is visible on your backup subnet, the traffic should take that route.

I had this situation just now with B&R 11.7
I modified the HOSTS file on my Exchange Server to point itself to the Storage Node's storage VLAN IP address

ex:
172.31.200.1 storageNodeHostName
172.31.200.1 storageNodeHostName.localDomain.local

Running a full job and I am observing traffic on the storage VLAN :)