Greetings Taifun! scanning the forums it seems like you are the right person to ask. Im making a simple radio app for myself and was wondering if it is possible to make a player run in the background even with the screen off using app inventor? maybe you even have an extension for it! im happy to pay.
yes, it seems to be possible, however I never did this myself...
for solutions please do a search in the community here or in the Kodular community
Taifun
Trying to push the limits! Snippets, Tutorials and Extensions from Pura Vida Apps by 
Taifun.
See here:
thank you. this was the solution i used.