Who can submit papers to peer-reviewed journals?

Anyone can submit a manuscript to a peer reviewed journal. The challenge that face persons without training in scientific writing through, for example, a PhD, is that the manuscript is probably far more likely to be rejected because of poor writing or other mistakes. The key issues for authoring a good paper is to have a good grip on the literature in the field, knowing the sources that should be referenced to provide the basis for the own work, to know how to write clearly, concisely and precisely, and to understand any specific publishing aspects of the field.

It is not rocket science but usually requires both good coaching and training. Approaching the authoring with care is therefore a good strategy.

Anybody can submit to almost any journal (there are some out there where some sort of existing membership is a pre-requisite, but they are rare exceptions).

There is, however, an additional barrier that an author has to overcome if they are not known and/or they are not affiliated with some reputable institution. In essence, peer review of a journal submission is attempting to evaluate credibility of the arguments presented in the paper. Authors who are known in the community that they are submitting to or who are coming from highly reputed institutions have an inherent advantage in that they already have some credibility simply through their reputation or affiliation. An author who does not have these advantages will naturally be faced with more skepticism about their statements, particularly when dealing with an experimental work where the paper cannot contain every relevant fragment of information about the work being reported.

What this means, in practice, is that when a submitted paper has flaws, an unknown and/or unaffiliated author is more likely to get rejected whereas a author drawing on prior credibility is more likely to get asked to make revisions. This might not be ideal, but pragmatically it is fairly reasonable: there is a lot of really bad stuff submitted to journals, and the quality of a submission is typically fairly well correlated with author and institution.

So how should an unknown and/or unaffiliated author go about publishing? First off, it's very useful to get feedback on pre-submission drafts from trusted colleagues, so that the initial submission can be as good as possible. Second, it's rare that any major development is contained within a single paper. Rather than trying to publish "the one big paper," one can build up credibility by publishing a sequence of manuscripts, starting in still-credible but less prestigious journals.

For example, if the work is about a general new principle, there could first be a paper proposing the principle and analyzing its implications, followed by another paper making experimental tests of some of those implications, followed by a bigger paper pulling it all together an demonstrating the general power of the principle with more diverse experiments. These are all perfectly reasonable papers---no LPU dishonesty needed, just an understanding that most significant ideas usually result in more than one journal paper worth of work, and some idea of how to segment the work sensibly.

My initial reaction to this question was that the only requirement for submitting a paper to a typical journal is that one has to be a human being. But then I remembered Shalosh B. Ekhad, which, in addition to numerous papers with human co-authors, also has a few solo papers.