Here is the query
from a in this._addresses where a.Street.Contains(street) || a.StreetAdditional.Contains(streetAdditional) select a).ToList(
I don't think SqlServer gave you a null exception. If it did, then this code is clearly not running though LinqToSql (as you've tagged the question).
string.Contains would be translated to sql's like, which has no trouble at all with null values.
like