@Repository
public class AccountDaoImpl implements AccountDao {
	@Autowired
	private SessionFactory session;

	@Override
	public List getAllAccounts() {
		return session.getCurrentSession().createQuery("from Account").list(); //здесь возвращается все правильно
	}

}

@Service
public class AccountServiceImpl implements AccountService {
	@Autowired
	private AccountDao accountDao;

	@Transactional
	public List getAllAccounts() {
		return accountDao.getAllAccounts();
	}
	
}

@Controller
public class ViewResultsController {
	@Autowired
	private AccountService accountService;

	@RequestMapping(value="/viewResults", method=RequestMethod.POST)
        public String setupForm(@ModelAttribute SearchParams searchParams, Map<String, Object> map) {

		map.put("searchParams", searchParams);
		map.put("accountList", accountService.getAllAccounts()); //здесь все работает
		accountService.getAllAccounts(); //здесь все работает
        
        return "viewResults";
    }	
}

@WebServlet("/AccountsServlet")
public class AccountsServlet extends HttpServlet {
	private static final long serialVersionUID = 1L;
	
	@Autowired
	private AccountService accountService;
	
	protected void doGet(HttpServletRequest request, HttpServletResponse response) throws ServletException, IOException {
			
		PrintWriter out = response.getWriter();			
		accountService.getAllAccounts(); //вот здесь не срабатывает, java.lang.NullPointerException	
		out.print("ok");
	}
}

@WebServlet("/AccountsServlet")
public class AccountsServlet extends HttpServlet {
	private static final long serialVersionUID = 1L;
	
	@Autowired
	private AccountService accountService;
		
	public void init(ServletConfig config) throws ServletException {
	    super.init(config);
	    SpringBeanAutowiringSupport.processInjectionBasedOnServletContext(this,
	      config.getServletContext());
	  }

	protected void doGet(HttpServletRequest request, HttpServletResponse response) throws ServletException, IOException {
			
		PrintWriter out = response.getWriter();
		accountService.getAllAccounts();	
		out.print("ok");
			
	}