This year, 5772, the last day of Pesach in Israel is on Friday. Is there any possible way that one could contrive to eat chometz on the shabbos immediately following Pesach?
I don't see why a gentile couldn't give you some on Shabas; you could even eat it if he hadn't made it for you. Of course, consult your rabbi for practical matters rather than relying on what you read here.
 But see the comments to this answer.
As long as it's Pesach (i.e. until Friday at dark), a Jew cannot own any kind of chametz. Once Pesach ends, Shabbos immediately begins. A Jew cannot use money to (re)purchase chametz, and can't cook/bake any new chametz.
For this reason, even in Israel, observant Jews will use matza for their shabbos "challah" on the shabbos immeditaely after Pesach (known as the 8th day of Pesach in the Diaspora).
For Ashkenazi Israeli Jews, there is one way to make a clear distinction between the food on the shabbos after Pesach, and Pesach itself.
The custom of refraining from kitnyios on Pesach only applies to eating it. Ashkenazi Jews may own, do commerce with, and benefit from kitnyios throughout Pesach.
So, buy some Pesach-certified kitnyios food on Chol HaMoed. Once shabbos starts, break out the rice, corn, beans, have a festive kitnyios meal!
For shabbos lunch, I highly recommend a salami sandwich, on matza, with MUSTARD (another kitnyios item).
Since this question only applies in Israel, I must disagree with msh210's suggestion. Here in Israel, there is a high probability that the fresh chametz product came into existence with at least some forbidden assistance from a Jew. Even if it came from an Arab-owned bakery that is certified kosher - that kosher supervision doesn't apply on Pesach! So, you can't eat it because it wasn't under kosher supervision at the time it was baked, so you have no way of knowing what exactly is in the chametz product.
Per the Yalkut Yosef 448:5
It is permitted to eat Chometz on a Shabbos which is immediately after Sheviyi Shel Pesach - there is no Muktza involved, and you if it was sold to the non Jew according to Halacha you may eat it on the day of Shabbos immediately after Sheviyi Shel Pesach. However you have to be extra careful not to take the chometz on Sheviyi Shel Pesach since then you will be prohibited to eat it since you have done Baal Yiro'eh U'Baal Yimotzei. (Yechave Daas 2 Siman 64, Yabia Omer Orach Chaim 9 Siman 46)