The Chicago Manual of Style (14th Ed.) provides the following commentary. I am personally not aware of any other major style guides (MLA, AP, Yale Law, etc.) that have any material difference in opinion.
Direct quotations must reproduce exactly not only the wording but the spelling, capitalization, and internal punctuation of the original, except that single quotation marks may be changed to double, and double to single as the situation prescribes, and commas or periods appearing outside the closing quotation mark may be moved inside.
The Chicago guide lists "A few [four] other changes" that are permissible, but these are largely not relevant to the question here (and I certainly do not want to overstep the boundaries of fair use in regard to Chicago's exacting description of common practice).
Chicago underscores the idea that differences between British and American English (and other dialects that do not need translation) should NOT be corrected by stating that, aside from "modern" sources that may be safely assumed to be in obvious and inconsequential error, "any idiosyncrasy of spelling should generally be preserved".
If you are really into 'splitting hairs,' an (authoritative) editor's conception of the anticipated audience may, nevertheless, trump the common practice. Most authorities would agree that it is permissible to alter quoted material to conform to audience expectations provided that an explicit notice of change is provided along with a clear explanation of what exactly has been changed.
For example, you could provide a note that states that "The spelling of all quotations from American sources has been regularized to conform to modern British standards." But, if you think about it, why not just produce your quotations exactly and provide a note that indicates the opposite, that "The spelling of all quotations from American sources has been reproduced exactly as originally published"?
I think that almost anybody should be able to see by this thought experiment why reproducing quotations exactly, to the farthest extent possible is generally the best policy.
Therefore: in the context that is provided in the question on this site, the quotation should be rendered exactly as the original, with no correction and (furthermore) without the appending of any "sic." notice.
NOTE: When quoting material exactly that includes typographical and grammatical errors that may be erroneously attributed to the sloppiness of the author or the printer, the parenthetical notice "(sic.)" is typically applied immediately subsequent to the quotation. Differences in dialect are NOT, however, "errors." Therefore, the use of "sic" in this instance would itself be erroneous.