I think this depends on a few things, but the general answer is 'no, you can not run exe files hosted in a SharePoint folder'.
Are you using SharePoint Online or SharePoint Server? Also, normally do your employees have to download the exe locally, then run it from their computer?
If yes, you could use Microsoft Graph API or SharePoint Framework (SPFx) to download the file, then use something like WebRun(installed on empolyees machine) to run the locally downloaded file.
WARNING - THAT CAN BE EXTREMELY UNSAFE TO THE Nth DEGREE!!!!!!
You do have the bonus of this being an internal tool only, however you might still want to worry about disgruntled employees modifying your exe which gets auto launched.... no bueno
You could also try using the SharePoint File Picker, however I think it'll only allow you to download files not run them. The github is for OneDrive, but it can also be used for SharePoint:
There is also a
File Browser and
File Handler
Microsoft Graph is confusing, but it's the preferred way of interacting with OneDrive and SharePoint
Another option might be to sync SharePoint and OneDrive... I'm not sure if it would actually open up more options for you as far as 'running an exe' goes, but you could create a shortcut from OneDrive
Tips:
If you're still searching for a way to 'run' or 'open' an exe file on a SharePoint server, we can try and narrow down your options so your not searching all over the place. A REST API most likely won't be your solution, this is because they're used for CRUD operations (Create, Read, Update, Delete)... while 'Read' may sound like it will open or run a file, it will actually either get the file contents or get file metadata. The only operation that could help is 'Create' and that's only if there is an endpoint to 'create job' or 'create action' which would schedule a task to be completed on the server. So the solution is probably not in a REST API, but there's also the Microsoft Graph API and this is more robust and can provide more fine-grained control over resource management since it isn't limited only to simple CRUD operations.