From this page you can download all Latin squares of orders up to 11 that have a transitive autotopism group. (The lists are the same if you ask instead that the paratopism group is transitive).
For orders 2,3,4,5,7,11 the only examples are the group tables.
If you want to know what format these files are in, it is my usual latin squares format.
There are only two examples in the above lists that have sharply transitive autotopism groups. Neither has a sharply transitive paratopism group. The examples are both of order 8. I've put the last in the file.