Does NATO need UN permission?
In my Opinion:
No they don't, but NATO is a defensive pact for NATO member countries. NATO cannot act as a grouping unless a NATO member country is attacked and then the individual member countries can still decide in which way they will assist with a response.
Bottom line is Ukraine is not a NATO member. The NATO members who are delivering military aid are doing it as individual countries too, not as part of a NATO decision. They may discuss assistance to Ukraine as a group in order to align what they supply, but they have to act individually as sovereign nations.
That said. NATO can act like they did during the break-up of Yugoslavia but were doing so under a UN resolution I believe.
And also, if Russia or Belarus does anything drastic like using Nuclear Weapons then NATO can declare that as an act of war against NATO member states as fall-out would most likely be detected over the many NATO countries surrounding Ukraine. They probably won't do a nuclear strike back at Russia, but they would most definitely use the opportunity to push the russians back and to eliminate their forces by overwhelming conventional weapons.