I've requested teachers in a couple of ways. At ds6's school we went in and asked for math acceleration and a specific teacher. The red carpet was rolled out.
Last year with dd8 we asked for a specific classmate who had requested a certain teacher. She has been with this teacher for 2 years now and next year will move on to another. For next year I mentioned it to the teacher we are hoping she will land with that we are hoping for her.
We've requested a type of teacher. Right before dd went to the TAG school we wrote a simple request asking that dd be placed with a teacher who can differentiate for students, who would be able to challenge dd, and who was gentle and nurturing. No idea what might have happened with that request as we didn't stay in the same school.
There is also a lot to be said for volunteering at the school. I volunteer in my spare time because I love it but the week before our meeting at ds' school I had run a fundraiser that made enough money to buy 2 classrooms all of the technology they needed for next year. I think that probably had a bit to do with how accommodating our principal was. She's a good person and a real advocate for children but I think my efforts didn't go unnoticed.