If you don't have all the required documentation when applying for council housing, you should still submit your application. Southend council suggests you do this to start the process, which may help avoid unnecessary delays or complications. They'll contact you later to request any additional documentation needed.
For a smooth application, make sure to include the essential documents such as proof of identity, proof of residence, and proof of income when you can. If you need more guidance or wish to apply, you can find more details here.