Try something like:
ps -efl | grep wget
if (( $RETVAl == 1 ))
#no running copy of wget found, so launch it
elif (( $RETVAL == 0 ))
#running copy of wget found
#grep returned "2", indicating an error => may be worthwhile to alert you somehow
Of course, you may need to fine-tune the "ps" command (ie only look for processes owned by your user, etc).
An alternative and commonly used approach is storing a lock-out file in /var/run.
In your case (non-root), you'll need root access to create a subdirectory under /var/run you can write too. /var/run itself is usually writable only by root.
Your code would then look something like this:
if [[ -f /var/run/your_dir/wget_lock_file ]]; then
#file exists, so wget is assumed to be running
#no lock file, so wget shouldn't be running
touch /var/run/your_dir/wget_lock_file; #create lock file, to lock out future wget calls