No, you absolutely can't use a cell phone in the air like that. It is illegal for good reason. Imagine how many cell towers it would light up, until it finally got high enough to not light up any. Also cell tower antennas are directional horizontal, so won't even work well when you are too high.
If you are going to do this via radio, and that is probably the only realistic option, then the bandwidth and power you will need will almost certainly require explicit licensing. Perhaps there are off the shelf solutions that have already done this, I don't know. In any case, this is not a do-it-yourself radio link project.
This is a long shot, but if you can put a array of IR LEDs on the bottom facing down, you might be able to rig up a passable receiver on the ground by putting a telescope in front of it. Keep in mind that IR refracts differently than visible light, so you probably want a reflector. This only works line of sight and it will require the telescope to accurately track the balloon.