I recently shifted from Engineering to Psychology (actually Engineering Education, but I am currently preparing a publication for an APA journal). I used LyX for my masters' thesis, and LaTeX is the only acceptable thesis format for some of the Engineering departments. When professors started requiring that I write my papers using APA6, I quickly tried to figure out how that could be done with LyX. I was not about to abandon my favorite document preparation system for nothing. I have found there are basically two routes you can go. There is an apa6e class and an apa6 class. The apa6e class is a minimal class that incorporates apa6 guidelines for formatting titles, but leaves references and things to the user to configure. The apa6 class is a bit more thorough. It is an updated version of the apa class and as such has three modes for preparing an article. It adheres fairly strictly to apa guidelines and the reference format is hard-coded. I recently had to borrow some code from this class and combine it with the Engineering dissertation template from my u to get a suitable dissertation format that makes both the college of Engineering, and the department (of Engineering Education) happy. I would also mention that for the publication I am preparing, we use structural equation diagrams that I prepare with TikZ and go right into the LyX document. I would hate to have to do those diagrams any other way.
So, while it is unfortunate that we do not have an official apa6 template with LyX, there are LaTeX users who do apa6 (like me). You can use that class with any document. To get it to work, I basically just copied the LyX layout file from the apa class and made a few modifications. Then, I downloaded the apa6 class file and used it. This works reasonably well. By the way, it is not the APA who decides whether to accept LaTeX or not, it is each journal individually. Frankly, I think this is sort of a chicken and the egg problem. If enough users wanted to submit with LaTeX, journals would accept it, but many won't bother until there are more LaTeX users. Also, I have found very few people in the social sciences who are even aware of LaTeX. Not closed minded, just unaware. They use MS Word and Endnote because they don't know that there are good/better (free) alternatives. Jacob