One way of identifying influential researchers in a field is to rank researchers by their h-index.
I found only a list of researchers ranked by h-index for those with very high h-indices ( >40). However, I cannot see the ranking researchers that are under this number.
Is there a way to get a complete list of researchers in a field classified by their h-index (e.g., h-index > 5)?
You can get a list of researchers who have Google Scholar profiles and have identified themselves with the field, ranked in order of total citation count by searching for researchers in that field on Google Scholar, via https://scholar.google.com.au/citations?hl=en&view_op=search_authors. That's not quite the same thing but will give you some of the answers you are looking for.
Supposing that the h-index could be used to measure researcher influence (which is debated), I would strongly suggest, in addition to @Significance, to use at least the three main sources: Web of Science, Scopus, and Google Scholar. In What do bibliometric indicators measure?, 2007, Kermarrec et al. showed the high disparity in coverage and results, and the sensitivity to errors (same names, duplications). This information can be complemented by Which h-index? – A comparison of WoS, Scopus and Google Scholar, 2008, Bar-Ilan. The bias observed with Google Scholar is highly dependent on the field.
Using a measure with three different tools (after some cleansing), and plotting the profiles on a 3D axis might help to avoid "missing" influential authors who lay off the main diagonal (data from Bar-Ilan's paper):
A pending issue is the difficulty to get authors assigned to a field. Journals can be associated to differents fields, and you could make a rule like: "an author who has published at least x papers in a journal from the field is taken into account". A drawback is that an author in that may have an high h-index mainly because of papers published in different fields. So although "influential" in your sense, he might as well not be influential at all in your field.
A solution could be to collect DOI of papers, and then use retrieval tools as the ones described in Scientometric/bibliometric data retrieval from a list of DOI.
