St. Thomas Aquinas' contingency proof and LDS cosmology The argument from contingency summarized by St. Thomas illustrates that in order for existence to be a reality there must be an &quot;Uncaused Cause&quot; that possesses existence in and of Himself.  Everything else that exist does so by way of created contingency.
See CONTINGENCY

According to Mormon theology, God the Father is a physical being of
&quot;flesh and bones.&quot;  Mormons identify him as the Biblical god Elohim.
Latter-day Saint leaders have also taught that God the Father was once a
mortal man who has completed the process of becoming an exalted
being. According to Joseph Smith, God &quot;once was a man like one of us
and…once dwelled on an earth the same as Jesus Christ himself did in the
flesh and like us.&quot;
According to Mormon scripture, &quot;the elements are eternal&quot;. This
means, according to Smith, that the elements are co-existent with God,
and &quot;they may be organized and reorganized, but not destroyed. They
had not beginning, and can have no end.&quot; This was elaborated by
Brigham Young, who said, &quot;God never made something out of nothing; it
is not in the economy or law of which the worlds were, are, or will exist.&quot;
Thus, Mormons deny ex nihilo creation and instead believe that God
created or &quot;organized&quot; the universe out of pre-existing elements.

See MORMON COSMOLOGY
How does LDS cosmology address or reason with St. Thomas' contingency proof? In other words, according to the LDS, who or what causes existence to exist?
The comments are quite right about an LDS viewpoint being that it hasn't been revealed yet as doctrine. 

However your question is asking how God exists or when the beginning of everything was. 

The problem is you are asking an infinite question with only the capability of a finite mind. Joseph Smith attempted to help our finite minds understand that of eternal or infinite aspects.


  "I take my ring from my finger and liken it unto the mind of man-the
  immortal part, because it has no beginning. Suppose you cut it in two;
  then it has a beginning and an end; but join it again, and it
  continues one eternal round. So with the spirit of man." - Joseph
  Smith, Teachings of Joseph Smith.


Another way to think of it is that there are levels of science that we don't understand, elements of physics yet to be discovered that is above our physical capabilities in this current state. I know that doesn't give you a concrete answer but hopefully it helps.