UN World Food Programme is the main air ops agency within the UN humanitarian family, all other agencies (UNHCR, UNICEF Etc) operate under their indirect supervision to a greater or lesser extent. UNWFP is HQ'd in Rome, where they also provide technical & operational input to UNHAS (UN Humanitarian Air Service) and the UNASU (UN Air Safety Unit) which all UN agencies must go through when contracting or operating a/c.
Most ramp tramps are either working for UNHAS/WFP or as Gerhard states, are working for civvie contractors. Its highly unlikely you'd get in through the UN web site, 99% of the people I know doing this got in through knowing someone already in, or were on the ground when the vacancy arose, its all a big network.
UNDPKO (Dept of Peace Keeping Operations) run the military side of things, staff are usually supplied by contributing nations to accompany "donated" air assets.