Hi Weasleley!im in the midst of doing a similar show. We're doing a cabaret using musical theater songs and original scenes and monologues. We selected the songs and then hired six professional playwrights to write the acting material.
The songs is fairly easy. MTI will rent you standalone scores and materials or you can buy sheet music from somewhere like music notes. The next step is to figure out whether ASCAP or BMI had those rights. They can give you a performance license for the musical material. I think it's called a small performance license. It's good for the year. If you want to save money. I'd find songs that are all with one of the two agencies. Then you only need one license.
As for scene work. This seems to be tricky. You'd have to get an agreement with each playwright. I think many of them would be amenable to the idea, but you'll have to search and find their representation, contact them yourself, and wait for a response. This is why we decided to engage playwrights directly for née material.