Per TO page 354, the Homing Missile works like a standard Artillery attack, with a few exceptions. One of those:
"Instead, the homing missile may only attack units that have been successfully designated by a friendly TAG unit in the turn of the missile’s arrival."
A TAG the firing unit has is per definition friendly, and you'll find no prohibition for a unit firing Artillery weapons and using other systems in the same turn.
So, yes, the firing unit can TAG for itself, but keep an eye on the rules on page 185 of TO.