iakar,
You don't need to own the server to run CuteChat. As long as your host provider allows you FTP the files, you can copy the files to your server.
No, the CuteChat don't run on access right now.
asp.net Chat http://cutesoft.net/ASP.NET+Chat/default.aspx
Web Messenger: http://cutesoft.net/Web-Messenger/default.aspx
asp.net wysiwyg editor: http://cutesoft.net/ASP.NET+WYSIWYG+Editor/default.aspx
asp wysiwyg html editor: http://cutesoft.net/ASP
asp.net Image Gallery: http://cutesoft.net/ASP.NET+Image+Gallery/default.aspx
Live Support: http://cutesoft.net/live-support/default.aspx