I actually ran into this exact same issue recently. I was able to redirect my applications' output to a file using 'nohup' and redirection of the ouput stream and error stream. I found 'nohup' was built into my rootfs by default, but you may need to enable it on your system depending on your particular configuration.
My inittab entry for my program ended up looking like this:
where '/usr/bin/myprogram' is my application binary in the rootfs, and '/home/root/myprogram.out' is the name of my log file.
Note, I played around with placing the log in various areas of the rootfs and found that some locations would not work either for permissions issues, or possibly due to when they are available to applications during the boot process. I'm not really sure. But placing it in the home directory worked fine.