Can we run a Web application without web.config file?

Yes, we can run an asp.net application without the WEB.CONFIG file. 

If we are not setting any configuration in WEB.CONFIG file then it will take MACHINE.CONFIG file for default configurations. This config file will automatically installed when your application getting executed. 

i.e MACHINE.CONFIG file contains defaults settings for all the asp.net web applications.

Because all the configuration settings will be available under MACHINE.CONFIG file by default these settings will be applied to all asp.net applications. 

Note:
Web application run without web.config but the run without debugging mode 
In 2005 The VS When we trying to run first time any application the 2005 VS gives us warning that want to run application without web.config file.
The MACHINE.CONFIG file will be automatically loaded when .net framework is installed.