There are web apps that do this already. The process itself is straightforward (at least for mathematics). An editor-in-chief and a group of associate editors (who are always top academics in related fields) accept publications, sometimes to their email addresses, other times through some automated system. The papers are then assigned to the appropriate associate editor who is handling that particular subfield. The paper is then sent to a referee (sometimes two or three, especially if the first review is not satisfactory). The referee will be an expert in the field capable of evaluating the work. The referee (after some months, or more) will return a written report. Part of the report will be for the editor, the other part for the author of the paper. Some journals also request that the reviewer fill out some additional details, such as scores on how important, correct, appropriate the paper is, etc. The decision is passed on to the editor and if the editor agrees the author is informed of the outcome, along with perhaps an edited version of the author comments. They may be required to amend some minor problems in the paper, rework it substantially and resubmit or it may just be rejected.
Publication itself can be complex. It often involves journal style files, professional typesetters, sending page proofs to the authors for approval, getting transfer of copyright forms signed, dealing with diagrams and so on. This side of things is the side that fewer mathematicians actually care about. Much of it is not relevant for web-only publications.