I would solve this problem with two passes of Voronoi diagrams:
First Pass: Region Partitioning
The first pass would use a somewhat sparse distribution of points (i.e. the distance between the points should be relatively large) in order to roughly divide the island into regions (see the note below regarding point generation). Next generate a Voronoi diagram based on these points. This will divide the island into polygonal regions around each point as shown below:
Second Pass: Border Randomization
Now that the island has been divided into regions, then next step is to 'rough up' the boundaries between them. To do so, generate a new layer of points using a more compact distribution of points (i.e. the distance between points should be small) and again use these points to create another Voronoi diagram. Next for each smaller region, assign it to a larger region by checking its 'seed' point. This will result in a more jagged boundaries between the larger subdivisions. Here's a close up of what it looks like with both Voronoi diagrams in place:
And here's that same area showing only the final boundaries:
Comments on Point Generation
Regarding point generation, I like using a Poisson disc distribution in order to get a relatively nice & even distribution of points. The other common option is to get a similarly even distribution is to use Lloyd's algorithm on a set of 'regular' random points. LLoyd's is easier to code, but can take some trial & error to determine how many passes are required to give the desired result.
One potential problem with this approach is that the first pass partitioning may generate some very small regions. If you don't want them in your final result, I would simply merge them with a random adjacent region.
The illustrations I provided happen to be raster images, but this technique also works with polygonal / vector representations as well.