While this article is about OpenBSD's floppy support, if you already had OpenBSD on the machine and working networking, upgrading it without any external devices is as easy as:

    ftp -o /bsd.rd https://cdn.openbsd.org/pub/OpenBSD/6.7/i386/bsd.rd

Then reboot and at the boot> prompt, just type "bsd.rd".

On 6.6 and newer, just type sysupgrade and that's it.