Web Service to fetch article citations
There seems to be no service or tool that let's you download articles in bulk. This is probably so because most journals do have zero tolerance for bulk/automated downloads. Automated downloads are some times (may be most of the times) an attempt to infringe the copyrights. 

For example, ACM Digital Library ToS has this restriction.


  Under no circumstances are the following actions permitted:
  
  
  Using scripts or spiders to automatically download articles or    harvest metadata from the ACM Digital Library. This activity is a serious violation of  ACM’s DL  usage policy and will result in the temporary or permanent termination of download rights for the subscribing institution. 
  


That being said, irrespective of the field of research, here are some services (free and paid) that offers you a list of citations / cited by articles (other than the journal's webpage for the article).


CrossRef (some services are paid ones)
Web of Science (Paid)
Google Scholar (Free)
NASA ADS (for astronomy and allied fields. Free)


While ACM provide free Cited by list, IEEE provides this for subscribers/members.

Also please note, from NASA ADS FAQ


  In addition, references may be incomplete due to our inability to
  match them with 100% accuracy (e.g. in press, private communications,
  author errors, some conference series, etc.). Anyone using the
  citations for analysis of publishing records should keep this in mind.


which is true for any citation / Cited by lists.