My friend makes her dog little doggie coats from remenants of Polar Fleece cause it doesn't frey and doesn't need hemming. has it sewn across the chest so it slips over their heads, open underneath with a wide strap/buckle around the back and belly, just behind the front legs. I wouldn't put a second strap like horse blankets have, because goats kicking around might get a leg caught in that, especially playful kids.
I think a croup strap for a goat would be good too, to keep it in place better... might get messy though. not sure how I'd do that though.
Polar fleece is slightly stretchy and very forgiving though.