Okay, I think I may have a solution for you.
In your sudoers file, add the following line:
sniper ALL=(reporter) NOPASSWD: ALL
Where "sniper" is your username. This means that the user "sniper" will be given permission on "ALL" hosts to run "ALL" commands as the user "reporter" without requiring a password. (More on this below.)
In order to run the command, you would type in the following line (when logged in as "sniper"):
sudo -u reporter -i /home/reporter/somescript
The -u argument tells sudo to execute the following command as the user "reporter." The -i argument tells sudo to log in as the user "reporter" when executing the command. "somescript" is a bash script in the directory /home/reporter.
Inside of somescript (or whatever you decide to name it), you can put the following two lines:
cd /path/to/directorywithscript && bash runwebserver.sh >> /dev/null 2>&1&
Also, be sure to "chmod 755 somescript" in order to make it executable. The reason to put this in a bash script is that it results in less typing for you. You can type the full line in otherwise, but work smarter not harder etc etc.
This is the quickest way to get things working the way you want, but it may not be the most ideal. For one, any command can be run as reporter by using the above command. In fact, the command "sudo -u reporter -i" (exactly as it is) will log you in as the user "reporter," which may be giving away too much access to reporter's account. However, this is necessary as full access is required to run any login scripts that reporter has, as well as any commands in the script files. There are probably other, more secure ways to do this, but it would require a different approach than what you're using.
In any case, it should do what you want it to do. Be sure to check out the Sudoer's Manual for more info:
[EDIT]Catkin's solution is closer to what you would want to do in a server environment, so if possible, use that solution first. The page wasn't reloaded from when I first opened it, so I didn't see his response before I posted mine.[/EDIT]