Meteors, or anything else for that matter… do not ‘catch on fire” when entering the atmosphere.
However ALL of them experience atmospheric entry heating and yes, some of them do vaporize into tiny little particles because of this heating. But it is not Burning as with a flame.
Meteorites entering the earth’s atmosphere at high speeds will heat up to red hot by the sheer friction with the air.
Since most meteors and meteorites are very rocky in composition there is little or no combustible material, the rock will mostly melt and flake off in pieces. from the heat and physical stress, as well as glow brightly in the sky (esp. at night).
Now you know that its not burning. Its the friction that causes the meteor to melt and to provide that friction, oxyen is not at all necessary. Other gases in the atmosphere can do that.
Some extra knowledge...
One other important thing to note is that oxygen is necessary for burning. One of the more commonplace fires that doesn't use oxygen (O2), are metal fires. Some metals, such as magnesium, will burn when heated in the presence of carbon dioxide, which is why a lot of welders will have specific fire extinguishers at their job site than the common CO2 extinguishers.
Interesting question though...good luck to the curious mind :)