my phone recently broke, and the screen when I touch it doesn't do anything. But other than that it works, I receive messages and what not and it shows up on the screen, and the buttons work too. I'm very concerned that I wont be able to save my photos. I tried syncing it to iTunes but it wont work unless I unlock it, which is not possible. And I'm positive I made an icloud that shares family stuff but I don't know to view it.
Tyson 's answer will probably work, but it would require that you enabled photo stream before your phone broke. If you didn't, you're kinda out of luck unless you've backed up your phone to your computer previously.
If you have backed it up before, you can use tools like iExplorer (free demo, $35) to read the iTunes backups. They do, however, make you purchase the software if you actually want to pull the photos out, but it may be worth it to some depending on how much they want their photos. Best of luck.