A simple Google web search for dramatic play sites will inform you of the many ways producers of school plays encourage legal application for permission to perform a school play.
The most important aspect of this process is that you contact the writer, his or her agent or publishing company directly. This will ensure that you know what is expected by them. Be sure you that you are fully happy with conditions before you go ahead producing the play. It would be very difficult for you later if the production suddenly comes unstuck and you need to change plays.
The author is the legal owner of a play and you need to acknowledge this. You will not only pay royalties but will be expected to acknowledge authorship at the end of the performance and on brochures and other printed material advertising that is available during the play.
The best way to get a feel for the situation is to write and produce a play of your own. Once you have been through this lengthy and time consuming process you will be in a better space from which to relate. You will now know how the author would feel about others using his or her own work without permission.
If you are a drama teacher, the chances are that you do have a play just waiting for the day you find an avenue through which you can express it. Consider writing the play for your school and then after 'crash testing' it - submitting it to a publication house. There is no better way for a playwright to hone his writing skills than to see his work performed. When you have produced the play, search the web for submission guidelines and write to the editor asking whether it is the right time of the year to submit and what is expected of you.
Here is an example of submission guidelines.
Do not be discouraged by the idea of royalties. They are not that hefty when you pay for them by charging a small admission fee. You will feel confident about the fact that you have contributed to the dramatic world. When the time comes for you to publish your own play, your rights will similarly be protected and you as an author will be respected and acknowledged.
The world of royalties is the world of ...."Do Unto Others."