Short stories in Spanish are available online at E-Stories.org or in hard copy via Amazon.com. E-Stories.org lets you find stories based on the language in which they're written in or the topics of the stories.
On E-Stories.org, click on the title of the story you want to read to have the text open on the page. The author is listed above the story.
Searching amazon.com for "Spanish short stories" yields a list of relevant books including "Short Stories in Spanish: New Penguin Parallel Text" edited by John L. King and "Spanish Short Stories 1" edited by Jean Franco. Both of these books received ratings of four stars or more on Amazon.com.