The proper name for a “story with photos”?
This is sometimes referred to as photojournalism or a photo-essay.
The links above are to Wikipedia, but here are definitions from Oxford English Dictionary:
Photo-essay: A series of thematically related photographs, usually supplemented or linked together by a narrative text or commentary; an essay, biography, etc., consisting of numerous photographs, with or without accompanying textual matter.
This definition is first cited in 1948.
The Atlantic Coast Exhibition by Eliot Elisofon, a Life Magazine photo essay, closes this week end.
- 1948 Portland (Maine) Press Herald 5 Nov. 15/5
Photojournalism: The use of photographs in journalism; journalism consisting primarily of photographs.
The first citation of this term is from 1938:
While Photo-Journalism is a comparatively new development in America, it has already taken quite an important position in Europe, where it developed during the past ten years.
- 1938 A. Eisenstaedt in W. D. Morgan & H. M. Lester Miniature Camera Wk. iv. 67/1
Your link points to what is colloquially called a "photoblog" or a "photolog". It is rooted in the concatenation of "photo" and "blog". However, it is not recognized as an official word by OED or Webster (yet).
Photoblog : Oxford Living Dictionaries
photoblog NOUN
A blog in which the postings are primarily in the form of photographs.
‘I'm helping them set up a photoblog which will hopefully give everyone a chance to see some of those amazing pictures’
Wikipedia has an entry.