It is publicly available, but it is also, um, in German, and I'm not sure how accessible it really is to the layperson? There's lots of abbreviations and stuff. But - okay, here's the link, and if you want to look at tablets, maybe it'd be best if you search by CTH number? That's Catalogue de textes hittites, and there's a list of all the numbers and what they mean here, but that's also in German, sorry. Anyway, not all the tablets have good pictures, but some of them really do, and if you click on the little camera picture in the leftmost column (if it's red that means there's a picture uploaded) you can see them.

The parent site is the Hethitologie Portal Mainz, which has a lot of other links, although again it's mostly in German and mostly designed for specialists. If you want a site in English for laypeople, though, you might try hittites.info? They're pretty decent, and they've got history and maps text translations and stuff.

Hope all that's helpful! It's always awesome when people are interested in the Hittites (the best of all ancient civilizations!).