Can "contemporary" be used for objects
Solution 1:
The sentence
Our business stands out from our contemporaries.
forces business and contemporaries to be compared as though they were the same class of object, when what you really wish to compare is your business and the business of your contemporaries. Therefore, to be clear and precise in modern English you should compare those things, as follows:
Our business stand out from the business of our contemporaries.
Or, in case you are using business as a count noun to denote that each is a business that does not partake of a common set of features, try
Our business stands out from the businesses of our contemporaries.
You can avoid repetition, if you wish, this way:
Our business stands out from that of our contemporaries
or
Our business stands out from those of our contemporaries.
Solution 2:
While contemporary is usually used for people (specifically as "his/her/their contemporaries"), it can also be applied to objects. To list a few examples found through the Corpus of Contemporary American English:
Motobecane was a leading brand. After fading away (along with most of its Euro contemporaries), the brand is now making a comeback. (Bicycling, 2002 (Apr), vol. 43, issue 4, pg. 27.)
In fact, the handset drops the removable battery altogether, in order to make the device waterproof like many of its contemporaries ("More info leaks out ahead of LG's G6 launch," TechCrunch, 2017 (17-01-26).)
She has compassion for and insight into the teenage experience and she treats it with an understanding that lifts the film above its contemporaries. (Review: 'Edge of Seventeen' enters rare teenage air." Detroit News, 2016 (16-11-17).)
Each of these examples shows that "its contemporaries" describes objects like the prior specified thing: the examples above involve Motobecane and contemporary brands, the device and contemporary devices, and the film and contemporary films. If you wanted to do this as well by focusing on contemporary businesses, you could write:
Our business stands out from its contemporaries.
Its is necessary to specify the object (the business). Alternatively, your current phrasing ("our contemporaries") specifies the contemporaries in relation to us, and could include your coworkers and colleagues. It's comprehensible but awkward. If you truly meant to specify the people in those businesses as your "competitors" or "contemporaries," then any of the rephrasings given by Robusto would accomplish that, e.g.:
Our business stands out from the businesses of our contemporaries.