If that is the arrangement then you are the administrator / admin for maintaining the site and if they want corrections, they need to ask you.
This should be part of the contract between you and the company.
If they want to alter things like people contact details, then that should be fairly easy to add an admin page for them to access.
If you pay for the server, then you are under no obligation to hand over any access to that server, unless it is in the contract.
Best option is to ask them what they want to have access to and if they are evasive, then they are possibly wanting to take on maintaining the site, if that is so, you could sell them the whole domain and rights to the site code, they have only paid you for hosting and the initial design.
Unless you have any specifics in writing about the terms of the contract, you will need to do your phishing carefully otherwise you may tip them off that you are on to them...
So the options are (from what I can see)
1. Offer to add in an admin page so that they can change details for things like contacts, offices, telephones, page cosmetics (css, images, banners) and profiles to give them some access which is likely the most they will need, chances are that they may want to be able to upload and manage files for distribution, this is where you could negotiate to produce this level of tool for them in the site. But you will need to ensure that they know that this development is not part of the original fee and it is new work, therefore you will need paying for it.
or you could
2. Sell the code and domain to them rights to them for a lump sum.