I just started playing the trombone not long ago, and I am curious how to do vibrato on trombones? I used to use my diaphragm when fooling around with vibratos, but was never sure how to correctly execute them.

lets see if i can remember this there are 2 diferent ways to produce vibrato. The slide vibrato often called a shake and the lip/mouth produced vibrato. and there are 2 school of thaught on which is better.
Lip vibrado is produced by changing the shape of your lips ever so slightly while playing this can be done by flexing muscles around your lips and face area slightly openiong and closing your mouth(eg your jaw moves) or by ajusting your air flow with out chainging volume. This vibrato is bad because it creats muscell memory and then makes it hard to produce a steady tone. Also it is bad because as you change the shape of your mouth you are screwing with your ombeshure.

The shake is created by ajusting your slide back and forth much like a violinist might do to there strings. this is bad because it creats a habit of "i dont really know where the positions are i just get close nd wiggle and it will be ok."

I personaly use both on diferent ocasions and find them both good for me. i recomend talking tou your instructor about what he recomends you use, maby dont use any at least for now.