A lot of users have trouble unlocking the coffee machine. To do it, you need to complete the 'Upgrading I' mission, indicated by the icon with a white shirt and a light blue cravat.

You then need to complete 'Spruce Up Your Cafe I' and 'Spruce Up Your Cafe II' which opens up now, indicated by a white shirt with a dark blue cravat.

You should then see a coffee cup appear on the right hand side of the screen. Select it and a crate will appear. Move it into position and then click it to open it.

As long as you have enough neighbors, it will open.