Updated based on The Amazing Spider-Man 2

No. The spiders would not have successfully given powers to just anyone.

In a video log, Richard Parker explains that his own DNA had been fused with that of the spiders. Only someone with shared DNA (such as Peter, or likely Ben Parker) would successfully gain powers from the spiders.

Original answer based on The Amazing Spider-Man

In-Movie answer: No, not just anyone would probably gain the powers.

Dr. Connors mentioned that every cross-species DNA merge resulted in death. Even with what should have been a stable formula, Connors body did not react to the formula the same way - changing him into a giant Lizard while Peter simply gained powers.

Not-Quite-In-Movie answer: There is probably something special about Peter.

There is a fantastic write-up about The Untold Story which was cut out of the movie. It discusses hints of deleted scenes that were in trailers and pieces them together with the final film to theorize the supposed "Untold Story".