vga.asm copies bytes directly to video memory in a raw manner, once it's in VGA mode. It only uses BIOS to switch into VGA mode ("mov ax, 0x13" and then "int 10h"), and back to text mode before rebooting when you quit. It does use BIOS to read the keyboard. It also reads bytes from sectors of the disk using BIOS rather than the OS when it starts up. The sound is a beep created with raw "OUT" commands on the data ports (for the PC speaker presumably).

In other words, it's using BIOS for input, but not drawing.