Overview:

When a user is recording an Inventory Monitoring Online or Offline, the following error message occurs: Insufficient access rights on cross-reference id.

Root Cause:

That is due to Territory Settings assigned to the user. 

As there is a Master-Detail relationship between Account and Inventory Monitoring, users assigned to that Territory need View and Edit permission.

Solution:

To enable View and Edit permission for the User's Territory: 

1. Navigate to Setup --> Manage Territories --> Territory Models.
2. Select the Territory Name assigned to the user. 
3. Select Edit. 
4. In Account Access section select: Users in this territory can view and edit Accounts assigned to this Territory.
5. Select Save. 
6. Clear the Veeva cache.

Related Documentation:

CRM Help Documentation: Inventory Monitoring