"beg permission" is correct and usable in written English.
It is typically used in polite requests, when one is asking for the permission of another. For example: "I beg your permission to leave the room early."
We were astonished to find that as EU citizens, we had to beg permission to travel across its borders.
Rolo was provided with a golf umbrella and towel, a drink at 15-minute intervals, and the opportunity to go to the toilet in the nearby bushes without having to beg permission first.
Things are so bad that the boss of VW in the U.S. flew to Germany to beg permission for incentives.
If we wanted to travel together he had to book several flights to consulates and embassies to attend in-person meetings and beg permission to travel to the host countries as they limited the number of Jamaican visitors.
Arion, as a last favour, begged permission to sing a song.
Initially Ji did not know why he was being no longer selected so eventually Poyet begged permission to explain.
In 1979, when accepting a lifetime achievement award from the American Film Institute, he begged permission to thank four people who had given him the most "constant collaboration".
The work space was so limited that the two begged permission from a neighboring building to carry materials through, and had other materials lifted over the top of the building with a crane.
For three years I have been trying to interest sponsors and researchers in a project that would address this annoying business in which a human respectfully begs permission from a computer's software to turn the machine on or off.
A little East of Jordan, Evangelists record, A Gymnast and an Angel Did wrestle long and hard Till morning touching mountain And Jacob, waxing strong, The Angel begged permission To Breakfast - to return!
It wasn't a matter of begging permission to visit.
