Off the top of my head I can't think of any specific applications to do this, but as you say it would be fairly simple to create so you may be able to cheaply commission someone to write it for you (you don't state the nature of your company: if you have an IT department then you may be able to second some resource from there to do it for less than getting external contractors in might cost (then again, some IT depts cost more than externals!).
The hardest part of the idea is access control: making sure users can only access the files that are intended for them (and by inference providing an interface to specify who the file you are uploading is for). Of course if the files are not sensitive then you may not need access control, in which case you don't need an application at all: just have a browsable directory on the web server and upload files as you would normally (SCP, SFTP, ...). If you only need very basic access control, you can use the "just use the web server" approach and create a password protected directory for each client and send the password and location to them by email.
As for ready-made web apps for this that are available for purchase (or for free), I can't think of any. You might get more responses on this if you specify your web server actually is (IIS, Apache on Windows, Apache on Linux, lighttp, ...) and what scripting languages are supported on it.
EDIT: Just reread your question and noticed the "send and receive" part. The "just user the web server" option would still work for this but not nearly as well (as you would have to give them accounts to send stuff to you and if using SCP/SFTP you'd need to ensure they had the relevant client application installed), though the rest of the answer is still relevant.