Does Appian allow Business users to model and execute workflow processes

Is this a licensing type question?

Appian /design does not care what the user account's role in the company is - as long as they are added to the Process Model Creator / Designers group they will be able to create applications, process models. 

"Executing" workflows from the business side does not require any elevated access other than Initiator (typically we grant Viewer) access to the models.

Note if you do add random business uses to create applications, this can become very messy from an environment perspective unless you can ensure they adhere to proper coding and design standards.  There can also be security concerns if you allow them to access business data for areas they typically would not have access to.  For these reasons, we do not let anyone outside of our Developers group design applications (we have been asked..). 

If you are new to Appian, you might be surprised how much a small seasoned group of dedicated developers can build and maintain for the organization.

A few more details on your use case might be helpful also.