By 1945, the Luftwaffe was just a shadow of its former self, and was not in any position to put up any real resistance in the air. While small-scale sorties were flown right up until the end of the war (Erich Hartmann got his last kill of the war on May 8, VE day), Operation Bodenplatte on New Year's day was the last time that the Luftwaffe really went up in force. Bodenplatte was a surprise attack accompanying the Battle of the Bulge to try to destroy as many Allied aircraft as possible on the ground. While it was successful in this regard, it was a bit of a Pyrrhic victory as the losses were nothing that the Allies couldn't replace, and were far offset by the loss of Luftwaffe aircraft and some of the last competent aircrew.
The Luftwaffe had been in decline since late 1943, depending on how you look at it, and after the last hurrah of Bodenplatte didn't do much else in the war of note. Allied strategic bombing of the German oil industry (no gas for training), the attritional losses of fighting those bombing raids and on the Russian front, and a lack of pilot rotation had by 1944 severely degraded German pilot quality. This, combined with a lack of fuel, crippled Luftwaffe operations, even if they had plenty of aircraft to use.
Functionally they still had a lot of planes.
http://en.wikipedia.org/wiki/Luftwaffe_serviceable_aircraft_strengths_%281940%E2%80%9345%29
However competent pilots and mechanics were rapidly dwindling thanks to losses in Russia and the efforts of the allies in the west. So in short it wasn't the supply that was the issue, it was the maintenance and a distinct lack of qualified pilots.