The phrase "immediately upon entering" is correct and usable in written English.
It is typically used to express that the indicated action began immediately or shortly after someone entered into a particular place. For example, "They were greeted with a loud applause immediately upon entering the theater."
Immediately upon entering their new home, the housemates synchronize their overwrought emotional cycles.
Bess Kalb, twenty-eight, died immediately upon entering Burning Man with her new boyfriend, Travis or Trevor.
You are confronted with this immediately upon entering "Circa 1986," where a Jeff Koons sculpture stands just inside the door.
He was lucky -- when he saw people he knew, it was immediately upon entering the store, so he would turn around and walk out.
I was surrounded by an army of hospital-like caretakers who used hand sanitizer immediately upon entering the apartment, ate in our kitchen, showered in the bathroom and slept with my mother in one of our two bedrooms.
I was handed my first dish of ice-cream immediately upon entering, a scoop of Brooklyn-based Blue Marble organic vanilla covered with Froot Loops and marshmallows (the ice-cream purveyor changes about weekly, but they're all of the small-batch, locally made variety).
This calls for sending one shopper into the fray while another immediately gets in line upon entering a store.
So I left, and took the subway downtown to McNally Jackson, where, upon entering, I immediately saw six books that I didn't know about or had forgotten I wanted to read.
Fortunately, upon entering, we immediately saw a woman seated at a desk, yawning so wide that we were certain she had not slept a full night since the tragedy.
Therefore, north is the most important direction in Peking opera, and performers will immediately move to "center north" upon entering the stage.
Upon entering, visitors are immediately struck by its serene beauty – highlights include 16th-century Italian artwork depicting important Catholic events, and the elaborate crypt of 4th-century bishop San Filastrio that was constructed using materials found in the Roman and Byzantine ruins of the original church.
