Reception and circulation in their broadest senses leave traces all over the historical record, but the richest medium for reception history is the periodical. Magazines and newspapers that take note of fiction are a good place to look for responses to particular stories or books as well broader discussion of writers or trends.
Big nineteenth-century periodicals databases have some coverage into the early twentieth century:
After that, things get trickier. Some major newspapers’ twentieth-century runs are available digitally in ProQuest Historical Newspapers, for example:
All of these, of course, regularly review books and cover literature from their distinctive positions. With some fiddling you can do a search across all of these at once from the ProQuest Database Select page. Many other magazines and periodicals are included there or in other big databases. As you get closer to the present, current-news databases start to capture some periodicals of interest. Start from the Libraries’ master database list or search the catalogue for particular titles.
One magazine is a particularly useful resource for US reception history: Publisher’s Weekly. We have access to a digitization of (almost) the whole run. Issues are served page-by-page (there’s no index to individual articles).
Gale’s Book Review Index is much less useful than one would hope but it is worth a try. Also from Gale is the Literature Resource Center, which is a grab-bag of digitized reference works and usually not very useful except as a basic biographical reference, but occasionally something interesting turns up.
JSTOR is focused on scholarly journals but of course scholarly reception is reception too, and often scholarship supplies information or references about circulation and reception.
Digitized versions of individual bound volumes of periodicals can be searched in HathiTrust if you can locate them. There is no simple way to search across the whole run of a periodical or across multiple texts.
But you can take the problem from the other end and do a full-text search of the whole HathiTrust database. That is the richest search of books as well as a way to catch the kinds of periodicals that ended up getting scanned as bound volumes. A full-text search of HathiTrust for a book title or author name is likely to yield many many false hits, but if you can find ways to narrow down (by publication date, etc.) you can often find likely possibilities. Then if you are lucky you will be able to see some or all of the book in question. HathiTrust and the increasingly tumbleweed-strewn Google Books overlap largely but not entirely, and Google will offer you previews of many copyright books that HathiTrust hides. For that matter, an ordinary Google web search often turns up things that no other technique will.
More specialized SF resources
(This list is largely the work of Suzanne Boswell.)
Pulp Magazines Project: We will make considerable use of this database of digitized early-twentieth-century pulps early on in the semester. When it comes to reception, try looking a few issues after the publication of a story to see whether reader letters or editorial comment registers a response.
The Internet Speculative Fiction Database: Remarkable bibliographic database of science fiction (aimed at fans and collectors). Particularly useful for tracing reprinting histories. Often links to cover images as well.
fanac.org: This site is devoted to the preservation and distribution of information about science fiction and science fiction fandom. It hosts a large archive of digitized fanzines, as well as reports on conventions. Authors’ appearances in the archive are indexed in the name cross-reference.
Locus magazine reviews: a major review outlet for SF, starting in 1968, with reviews available online from 1997. Scans of earlier reviews ought to be requestable by interlibrary loan.
Fanlore: a wiki about fandom, written by fans; not exclusively about SF. Significant coverage of SF awards (Hugos, Nebulas), fandom controversies old and new, etc.
File 770: a long-running fanzine which Suzanne suggests as a particular rich resource for contemporary fandom.
As in all digitized searching, it is very important not to overinterpret the presence or absence of search results in itself. You need some kind of comparative baseline in order to assert that something was popular or was ignored in some segment of the field.