I created a Community (Napili Template) whose users are the same as the contacts of my accounts on Salesforce: they are (paying) members of my organization. I would like to enable them to get access and update (some of the) fields of their account (account object of Salesforce). Any idea of how I could implement that?
Have you tried to expose the Account object to them yet? You can open that Object to them and give them the correct Field Level Security so they only have access to what you want them to edit/see
You can use sharing sets to control access to their account specifically.
If you go to Setup - Communities - Community Settings you can create a new sharing set. Sharing sets let you use attributes off the user record and determine sharing of records based on those attributes.
E.G. The User.AccountId matches the Id of the Account record, then they have Read/Write access to that record vs just read access. You can do the same with their contact and other objects as well.