More detail would be helpful. Is this a website running on the NAS or is this a remote webserver that the NAS is mounting over ftp to? ie. you push files to the NAS and they get pushed up to webserver?
I'm guessing the latter and if that's the case, synology's UI does not have this option. There's no 'ftp' client on the command line either.
Check with your website provider, is ftp the only option? if SSH/SFTP is open you could potentially rsync the files over ssh from synology.
ie. https://www.digitalocean.com/community/articles/how-to-copy-files-with-rsync-over-ssh
Edit - Thanks for clarifying.
As Michael Hampton has said, ftp is very poor for file locking. So with these restrictions in mind.
- Define a share on Synology that is staging area for publishing from. Don't edit from the staging area, do that elsewhere, ideally in a version control tool, then checkout to staging.
- Install python on synology. In the web gui python 2.7 is available to install from the package manager.
- In command line of NAS, deploy the python based ftp script in this other answer https://stackoverflow.com/a/5664673/998808 - Its well documented with parameters for walking the filesystem and remote directory.
- Optional - Setup Cron to execute the script at particular times (say overnight) Instructions for cron on synology can be found here. http://jimmybonney.com/articles/manage_crontab_synology/
Note - for some reason all cron tasks in synology are executed as root, so when deploying the script you can put it pretty much anywhere but i'd suggest creating an upload user on the NAS and keep the ftp script in it's homedir. That way the script can be edited without dropping into the command line.
Edit X2 - As you've mentioned setting up GIT on another box, i'll point out that a GIT server is an available package on synology and there's various articles on post-commit scripts. This means you could do better than step 4 above by defining a GIT server that calls the python script post-commit and automagically updates the files across the webserver. Sorry i haven't more detail on this. I've not gone down the line of installing git on synology yet.