Teaching Citation: Are there any Journals which Fully Apply APA, MLA or Chicago Citation Rules
I believe all the journals published by the APA request strict adherence to the APA style manual. Similiary, PMLA, one of the two journals published by the MLA, asks for submissions in MLA style. University of Chicago Press publishes a lot of journals, but the first one I found asks for submissions in Chicago Style.

My experience with APA journals is that they allow for fair amount of flexibility in the submission format. I doubt any journal will reject submissions that use italics instead of bold or a comma instead of a period. They will likely fix these issues during copy editing,

As for the comments on the questions that just use Endnote or BibTeX/Biblatex to handle it, the problem is that most of these style guides are not designed to be programmatically implemented. It took a lot of work to get biblatex to fully implement APA 6 and there is no BibTeX version that is fully compatible.