Nataya, I looked into one local resource for you though it appears they provide direct rental aid rather than a deposit. Where I think it is a reasonable match is that they specialize in helping people qualify for a rental unit when their finances (security deposit, credit rating, income) do not meet the minimum landlord requirements.
Share & Care House is located in Tacoma on McKinley Avenue. You can call them at 253-564-4194 with questions.