Its not a great idea to have a web api call an executable. There are a number of issues you have to consider.
We are basically exposing the command line to the web. Any flaw in the way we start the console application might allow a malicious user to run some unexpected command
2: Running out of threads
With a web app we expect each request to be answered promptly, but we dont know how long the console app might run for. We could end up running out of resources
3: Permission issues
By default the console app will run as the same user that the app pool is running as. Will it have enough permissions? will it have too many permissions? you mught need impersonate a different user when you execute the app.
4: Will it close?
Not so much of a problem with a console app, but try firing up something like excel and watch your memory disappear
Your best option is to take the business logic out of the console application and use it directly as a library. But if you dont have the source code this might not be possible.
Look for alternate products which offer an api or library
If you really have to run console apps though, it can be done. Just be extra careful