Why is Rogue One allowed to take off from Yavin IV?

Question

Operation security at Yavin IV should be maintained at all times based on the importance of individuals visiting and taking part in the rebellion. Movements on and off the planet would be of utmost importance to monitor as any Imperial spy making off with a spaceship would be devastating to their operation. Here a former Imperial pilot operating an Imperial vessel taking off from the rebel base despite failing call sign check, authorization check, and clearance check.

Even the Rebel Sentry fails to take action when it launches. Standard procedure appears to be that the sentry scans all inbound and outbound ships, the Jedha outbound ship was monitored by the sentry on duty, but the Rogue One ship was not expected to take off.

Here is what a normal sentry action should look like:

Why is Rogue One allowed out of the atmosphere and particularly out of the solar system?

Answer 1

Basically, Rebels were disorganized and didn't know what they were doing.

From the novelization:

“Cargo shuttle, we have a pushback request here. Read back, please: Request denied. You are not cleared for takeoff.”
Bodhi winced at the comm and peered out the viewport at the tarmac. ...
“Yes, yes we are cleared,” he said. “Affirmative. Requesting a recheck.”
...
“I’m not seeing this request here,” the comm said.
...
“Are you sure everything’s been processed?” Bodhi tried. “Someone should’ve authorized it by now.”
...
“What’s your call sign?” the voice on the comm asked.
“Yes, we have it…” Just take off! “It’s, ah—”
Think, Bodhi. Give them something. Give them anything.
If you give them something, they might not shoot.
“—call sign Rogue. Rogue One.”
He transferred power to the thrusters, felt the familiar wobble of a cargo shuttle taking off under his control. The officer on the other end of the comm was squawking at him. Bodhi ignored it.
“Rogue One,” he declared, “pulling away!”

So, they basically were trying to figure out what to do, and didn't so much "allow" the shuttle to leave as didn't really have any way of stopping it (and it was theirs, so it's not like they could have started shooting at it without authorization).

Answer 2

The implication, according to the film's junior novelisation is that Bodhi's clever ruse was sufficient to buy them the time needed to take off before anyone in ground control was able to do anything about it. On top of that, he took off at an inadvisable speed, preventing them from doing much before the ship had attained orbit.

“Read back, please. Request denied. You are not cleared for takeoff.” Bodhi grimaced and then steeled himself to respond. “Yes, yes we are. Affirmative.” He hesitated for a moment. “Requesting a recheck.”
Rather than waiting for a response, Bodhi fired up the shuttle’s engines. The rest of the people in the shuttle—Jyn included—belted themselves in. It was either going to be a very short trip or a very fast takeoff.
“I’m not seeing this…request here,” the person on the other side of the comm said. “What’s your call sign?”
Bodhi glanced around, panicked. He had no idea what to say. “Yes, we have it. It’s, uh…Um…”
Bodhi looked to Jyn, but she just shrugged at him. He said the first thing that came into his mind. “Rogue! Rogue One.”
“There is no ‘Rogue One,’” the person on the comm responded.
“There is now,” K-2 said.
Bodhi gunned the engines and took the shuttle into the air.
“Rogue One pulling away,” he said.
An instant later, they were gone.

_{^{Rogue One: A Star Wars Story – A Junior Novel}}

Answer 3

They weren't. That was kind of the point of that conversation with traffic control.

Perhaps you're asking why didn't they get shot down? The answer to that is, why would they? The point of air-traffic control is typically to organize traffic to keep things safe. Without them, you'd be liable to have aircraft crashing into each other right and left. A person who ignores them and just does what they want is liable to have sanctions applied against them, like having their license pulled. Sometimes. But it would have to be really extreme circumstances to actually bother to shoot them down.

Now they could have just completely ignored air-traffic control. But informing them of their intentions to take off at least gave ATC enough of a heads-up to reroute other traffic to avoid collisions. They may have been rebels twice over, but they weren't sociopaths.