Unfortunately, neither of your proposed titles works correctly. "Favorite Vintage Life Photo's" leaves the possessive hanging: there is nothing for the photos to possess. And "Favorite Vintage Life's Photos" implies that the organization is called Favorite Vintage Life.
What you want is to use attributive nouns properly. Nouns used attributively qualify something about the following noun, a lot like an adjective, and can be more free-form rather than being limited to possession. Here, you're already using an attributive noun phrase almost correctly: "Vintage Life" can qualify "Photos" to mean "photos that are of Vintage Life in some sense". So you can just drop the apostrophe and you're fine. ("Favorite" continues to describe the entire following phrase, "Vintage Life Photos", so the title as a whole refers only to the photos that are of or from or by Vintage Life that are also favorites.)
Alternatively, you can rephrase to use possessives. In this case, you need to rearrange "favorite" so it qualifies the right words. It's not Vintage Life that's the favorite here, but Photos. So "Vintage Life's Favorite Photos" does the job just fine. It does have the quirk that, rather than describing your favorite photos from Vintage Life, it appears to describe the favorites that Vintage Life has selected for itself, which may or may not be the impression you want to give.