Yes, copyright law applies to all organisations and businesses, including not-for-profit organisations, members clubs, public sector organisations and charities.
So when you are showing movies and TV programmes publicly either to a group of the general public or a closed group of members or supporters then you still do need an MPLC licence as a charity or a non-profit organisation.
But don’t worry, licences are very affordable, easy to buy and can be in place immediately.