I think there are two possible fixes here:
First,
http://localhost:3000/needs/3/applicants.1
should probably read
http://localhost:3000/needs/3/applicants/1
The error is in this line:
<%= link_to 'Accept Applicant', need_applicants_path(applicant.need_id, applicant.id), :method => :put, :action => "update", :applicant => {:acceptance => true} %>
where...
need_applicants_path(applicant.need_id, applicant.id)
You can try passing in two instance objects like so:
need_applicants_path(Need.find(applicant.need_id), applicant)
Second, another possible solution is to explicitly set the PUT path in your routes.
In your config/routes.rb add the line
put 'need/:need_id/applicant/:id/update
then run
rake routes
and see what the PUT path is