The phrase "take up position" is correct and usable in written English.
You can use it to refer to someone assuming a place or role that has been designated for them. For example, "The soldiers took up position as instructed."
"I'll take up position here by the entrance," Mr. Davis was saying at our first stop.
Cells of the immune system take up position at vulnerable points of the body, ready to deal with injury and infection.
The two officers who have spent the last hour with him take up position on either side of the condemned man.
They are paid a lot less than presenters who may arrive just in time to read the running order, check their scripts and take up position in front of the cameras.
It took the Obama administration a few weeks to bow to that political reality and go from trumpeting the border as more secure than it had ever been to ordering National Guard troops to take up position there -- most of them in Arizona, Mr. Obama assured Ms. Brewer in a private meeting -- because it was not secure enough.
Cavendish and Gaviria had agreed before the British Tour started that the Colombian – who dominated the world omnium championship on the Paris velodrome in February – would get his chance on occasions and in the final kilometres of the 217km stage four Cavendish gestured to him to take up position in front.
He watched them take up positions and shoot.
The lookout sounded the alarm, and Menoyo ordered everyone to take up positions around the camp.
The government sent armored vehicles to take up positions and keep order there.
The rebels take up positions in farmhouses scattered across the hillside and shoot at the police.
Reinforcements arrived to take up positions on the bridge as automatic gunfire sounded.
