Can Alfred work without Spotlight enabled?
Alfred relies on the same OS X metadata index as Spotlight. As such, it's important for you to let Spotlight run on your Mac to maintain an index of the files on it.
Turning off Spotlight indexing will still allow you to use Alfred as a web launcher but you'll no longer be able to find files on your Mac.
Take a look at the Indexing Troubleshooting page for help with indexing issues.