"much" is an unqualified and unjustified word here. It definitely happens but this would at most affect a tiny fraction of donor money.

Many of the NGOs have strict no-bribery policies, else they would not receive support from bodies like the EU (which is the biggest humanitarian donor on the planet).

In some cases the choice may be between "letting people starve" and "feeding people but the local warlord extracts some benefits" but these are rare and only the worst crisis contexts (think South Sudan, DRC).