It likely got cut.
From the trailer, at 1m22s we see a whole army/battalion of mounted goats.
But in the movie, when the Lord of the Iron Hills' Dwarf Army shows up, they are purely unmounted infantry. Only Durin is mounted on a Warthog (Warhog). If you squint at 3m19, you can make out four others that may be mounted as well, likely the generals for four battalions, when the forward Infantry breaks off to meet the newly arrived Orc Army.
The goats that Thorin and co use to attack the Orc commander do come out of nowhere as far as we know. It's commonly mentioned online as not being shown on screen before hand. Either it's corrected in the dvd extended edition, or never made it past the cutting room floor.
If the four blurry generals were riding something, then we can assume those are the ones that Thorin took. Or the goats were kept back from the main battlefield. Logically, an army will need support, and mounts are perfect for carrying large amounts of support material, healing supplies, arms, tents, etc. They may have been over the hill until called.
In the books, the Dwarves are not described as having goat mounts. It's purely a (neat and logical) movie addition. The entire fight sequence was changed.