... Laz was to use FTPSend - but that unit is no longer in Laz (if it every was) ...
It is a unit of the
synapse project. see also
github.
You can install it in Lazarus using OPM.
Other than that, and perhaps you do not realize, your hints are vague at best. You have a webserver running so you ought to now what you run there and how to access. There are literally dozens of methods to 'upload' files to a webserver ranging from sending e-mails, remote shell access to running a dedicated service. Each method requires another approach. As an example, this forum software allows for attaching files that are being uploaded when you press the send message button. In order to let that work a http server is running on the webserver, along with some scripts that know what to do with those attached files when a user presses that send message link.
fphttpclient can be used in situations where a service runs on your website that actually accepts files (and know what to do with it, in case of 'sending' files). As long as you do not have anything hosted in that department the use of fphttpclient will be in vain.
Since you mentioned being able to access the server with an ftp client that means that your hosting provider at least grants ftp access, so the focus should be on that, Plenty of examples available either provided by synapse or here in the forums (and synapse is only one of several options) but.... it is important to know whether the server should be accessed using sftp or ftps as the actual implementation depends on it..