You may see in the table there are columns about rights holders. This is because in the world of performing arts another wrinkle is added for performing groups: performance rights.
In addition for paying for the sheet music for a show, directors are expected to pay for performance rights of their work. These rights come in either "small rights" or "grand rights"
Small rights - these are paid for use of the music in a cabaret setting only. You are not allowed to use any dialogue or choreography "evocative of the original show."
Grand rights - these are paid for performance of the show in its entirety. Everything should be done as intended. Any cuts, line changes, gender swaps, etc. must be negotiated with a representative from the rights holder.
As you can see, neither of these are very conducive for performance of scenes. Either you stand in place and sing the music, or you do the show in its entirety.
The grand rights for shows are mainly held by MTI and Concord Theatrics, and there exists ways to perform scenes of their shows under special circumstances.
MTI - if you are a member of the Iowa High School Speech Association, there exists an agreement to rent shows for a net total of $20 a scene. Further information can be found here.
Concord - for those not in Iowa, there exists (for lack of a better term) a loophole in their rights. According to concord you are allowed to perform what they call "cuttings" of a show provided it fits the following criteria:
1. 10 Minutes or Less
2. Performed in a festival setting
3. Performed for an invited audience of adjudicators
All scenes shown on this website follow the first of these criteria. The second is quite vague. A festival simply means multiple groups are sharing art for each other aka have more than one scene. Speaking to the "invited audience" section of number 3, there should be no public advertising for this event. Another option is to give each kid a set amount of tickets to ensure a truly invited audience. The final criteria, an audience of adjudicators, simply means the audience has to be judging what’s on stage. I promise this is already happening, but if you would like to cover your bases you can let audience members write comments over the scenes. It’s up to you how to interpret these stipulations, but as long as you reasonably meet the criteria, you have free reign to cut, paste, talk, dance, and generally just go ham on a scene.