I dont think the site is down, but it surely takes a while to load (the pdf is 30 meg).
The problem is that the Soviets were reorganizing their army in 1940 but Barbarossa forced them to proceed to another reorganization shortly after. So you have a TOE for 1940 and another for 1941 but it is difficult to know what equipment they really had in practice and what was only paper strength.
Airborne Brigade 1940 :
Strength : 3000 men and 11 tanks
It had the following elements :
1/ A parachute group with 2 battalions (546 men each). Each battalion had 3 companies of 141 men; and each company had a 50mm mortar platoon. On top of this they also had a signal platoon, a sapper/demolition platoon and a small recon platoon (37 men) and possibly a company with motorcycles/bicycles.
2/ A glider group with the same organization as the parachute group.
3/ An airlanded group with the heavy weapons consisting of :
1 mortar company (9x82mm)
1 aa company (12 heavy aa mgs)
1 tank company (11 T-40 or T-38)
1 arty battalion with a battery of 4x45mm and a battery of 4x76mm
Then in 1941 the new airborne brigade TOE seems to get rid of the gliders, increase the number of light tanks, reinforce the battalions and equip them with flamethrowers (1 platoon/battalion )
Airborne Brigade 1941 :
4 parachute battalions (678 men each)
1 tank battalion (50 T-37 later reduced to 32)
1 brigade arty with 6x76mm, 12x45mm and 6x82mm
1 recon company (bicycles)
1 aa mg company
So yes there was a tank battalion, well at least in theory. In practice it seems that although the airborne corps were at full personel strength, they lacked equipment especially tanks and radio. Besides it was not until mid 1942 that the reorganization was complete.