You can add starting your application into /etc/rc.local file.
Alternativelly you can place your script into /etc/init.d
root@imx8qmmek:/etc/init.d# cat test.sh
#!/bin/sh
echo test > /file.txt
root@imx8qmmek:/etc/init.d# ls -l test.sh
-rwxr-xr-x 1 root root 22 May 25 18:50 test.sh
root@imx8qmmek:/etc/init.d#
and add it into exact runlevel:
root@imx8qmmek:/etc/init.d# runlevel
N 3
root@imx8qmmek:/etc/init.d# cd /etc/rc3.d/
root@imx8qmmek:/etc/rc3.d# ln -s ../init.d/test.sh S50test.sh
root@imx8qmmek:/etc/rc3.d#
Is it what you are asking for?
Or you need the script to run as some user other than root?
In this case you can use "su" command in the script and point exact user.
I mean something like that:
# su the_user -c /path/to/the/command