I have received the following error message from a process Model which is not ch

There are a couple of options depending on your situation:
1. If it is a user task, following the instructions in the link provided by Stefan, you can manually reassign the task to an active user. You will also need to make sure the process model is not "hard coded" to the inactive user.
2. If it is a script task, you will need to edit the specific process instance that is generating the error and from the Assignments tab change the "Run As" option to run the node as a different user, save the changes, exit Edit mode, and then from Monitoring mode, restart that node.
3. If the that fails, you may have to modify process instance security using a smart service in the "Process Model Utilities" component that allows you edit process instance security properties.

Hope this helps

I wonder if you could also check the security on the datastore.  It's possible that its permissions are set to be too restrictive; from what I read Appian always defaults to only System Admins being able to read and write to datastores (other responders please correct me if I'm wrong). 

You can "run as" someone who does have the proper rights in this process model, but you'll have to keep doing it over and over every time you need to use it in future development, or you could just update permissions properly once on the datastore itself.

Provided, of course, that this is what the issue even is.

I wonder if you could also check the security on the datastore.  It's possible that its permissions are set to be too restrictive; from what I read Appian always defaults to only System Admins being able to read and write to datastores (other responders please correct me if I'm wrong). 

You can "run as" someone who does have the proper rights in this process model, but you'll have to keep doing it over and over every time you need to use it in future development, or you could just update permissions properly once on the datastore itself.

Provided, of course, that this is what the issue even is.