The weird and RAW answer should be yes, but it's probably not.
Before anything, I must say that I'm genuinely amazed and surprised by your thought process and found your question funny.
A figurine of wondrous power is a statuette of a beast small enough to fit in a pocket. If you use an action to speak the Command Word and throw the figurine to a point on the ground within 60 feet of you, the figurine becomes a living creature.
If it is a living creature it has it's basic life functions and mammals have the capacity to produce milk.
Unfortunately for you, only a portion of mammals can produce milk, that portion are the lactant female representatives of the species. I'll not lecture anyone in biology here, but I believe you know the prerequisites to reach that condition, right?
So, it's impossible?
I never said that. It's just highly unlikely. It becomes a living creature resembling the actual figurine. Then, if the artisan made a lactant female figurine of a goat, you have a lactant female goat and there's your milk.
My goat is a male, then I won't have milk.
Your goat still becomes a living creature with normal (and extraordinary) living functions. You can buy the females and have a breeding program of your own.
But yeah, unfortunately you most likely won't have your safe everyday goat milk in the middle of a dungeon.