The problem is that even though the Xbox 360 has got USB connectors, you can't use them to put your shots on an USB stick.
The only way you can get your pictures on a computer, is by posting them online on your Forza Horizon storefront in the game. Afterwards, you can download them by logging in on the official Forza website.
So unfortunately you really do need Xbox Live. (But I'm not sure if you need a Gold level membership to do this, maybe you can do this with a free Silver level account as well. Can somebody clear that up please?)