There is no single organisation or website in Essex.
Some local authorities ( Brentwood, Broxbourne, Chelmsford, East Herts, Epping Forest and Uttlesford) make council and/or housing association properties in the relevant parts of Essex and Hertfordshire available through the Home Option scheme
https://www.homeoption.org/default.aspx
but you still need to register with at least one of the local authorities first. (You're then automatically entered into the Home Option scheme).
Individual housing associations have 'nomination agreements' with local authorities, meaning that rented housing is only made available to people who've been put forward by those authorities. So you're going to have to do things the hard way and register with all of the local authorities within your areas of interest.
Constitutionally, Southend isn't part of Essex. (i.e. it's not within the control of by County Council). The city is a unitary authority. Their On the Move website is here:
https://www.southend-on-the-move.org.uk
Thurrock (which is the area around Grays, Purfleet, Tilbury, etc) is also a unitary authority. Their website is here:
https://www.thurrock.gov.uk/finding-somewhere-to-live/applying-for-council-housing
The websites that you'll need within 'proper' Essex (i.e. where Essex County Council has control) are as follows:
Basildon:
https://www.basildonchoice.org.uk/
Braintree, Colchester and Maldon (as well as parts of Suffolk, including the area that I live in):
https://www.gatewaytohomechoice.org.uk/
(See also here
https://www.braintree.gov.uk/housing/apply-housing, here
https://cbhomes.org.uk/find-a-home/apply-for-housing and here
https://www.maldon.gov.uk/info/20054/housing/9616/first_steps )
Castle Point (which is the area around Southend, such as Hadleigh and Canvey):
https://www.castlepointhomechoice.org.uk
Chelmsford:
https://www.chelmsford.gov.uk/housing/looking-for-a-home/joining-our-housing-register
Epping Forest (Epping, Chigwell, Chipping Ongar, etc):
https://www.eppingforestdc.gov.uk/housing/council-housing/apply-for-council-housing
Harlow:
https://www.harlow.gov.uk/housing-and-garages/council-housing/apply-council-housing
Rochford:
https://www.rochford.gov.uk/housing/housing-register
Tendring (Clacton and the surrounding area):
https://www.tendringdc.gov.uk/housing/housing-options-advice/apply-housing
Uttlesford (Great Dunmow, Saffron Walden, etc):
https://www.uttlesford.gov.uk/article/5308/I-want-to-apply-for-a-Council-or-Housing-Association-Home