Algorithm to generate all planar graphs
One approach would be to enumerate all graphs of a given size, then test each one to see if it is planar and filter out the non-planar ones.  This might work acceptably if you only want very small graphs.

Brendan McKay has a collection that enumerates all non-isomorphic planar graphs of size up to 11, which you could download and use directly.  There are about 16 million different planar graphs of size up to 11, so if you're lucky you might not need to consider anything larger and that collection will be adequate.