They rent crampons and ice axes, and sell them, as well as carabiners, harnesses, boots, tents, and other gear. There is a large climbing community in this city. You would expect to find every kind of community in a city of 35 million, and that's the case. There is also a speleological society, and Mexico offers some awesome caving possibilities.
Across Lazaro Cardenas on Articulo 123, and up a block at Calle Lopez #26 (between Independencia/16 de Septiembre and Articulo 123) is Deportes Rube, with comparable selection and rentals. Both places open daily about 11am.
While waiting for the stores to open, why not visit El Moro "Churreria", a 24 hour rendezvous for hot chocolate and churros. Churros are basically an "extruded" doughnut- dough, deep fried, covered in sugar; just what you would expect. It's on the east side of Lazaro Cardenas, between Carranza and Ayuntamiento. You really have to walk around the area, because even this skillfully reproduced map isn't too accurate. A couple of excellent cafes are on the corner of Ayuntamiento and Lopez. Generally, the best coffee in Mexico is exported. Thankfully here is an exception to the rule!
Also, don't pass up the opportunity to visit the Art Deco Belles Artes for Diego Rivera and Orozco murals, and the incomparable Ballet Folklorico. At the far end of the adjoining park Alemeda is the Diego Rivera gallery, housing his signature composition of a Sunday gathering in the park "Sueno de una tarde dominical in la Alemeda Central".