It's very unlikely that one of the big-name, prestigious journals will accept a submission from an undergraduate. However, many universities have specific journals that are open to submissions from undergraduates, such as the Iowa Historical Review for history undergraduates at the University of Iowa. If your university has anything like that, I would recommend submitting to that kind of journal. If you just want to get your name out there and show that you can do historical research and formulate a historical argument, this should do the job just fine.

Another option is to wait on publishing and further develop your topic into a full-fledged PhD thesis (which is basically a book, or at least a precursor to a book) as part of your doctorate program. Or, if you feel it's not possible to elaborate on your specific topic to that extent, and you think an article would be the best format, you can wait until you're actually in a PhD program (or have completed a PhD) and then submit your article to a major journal. It really depends on what you want to do.

Seems like a few things are going on here. First, think about where you're at and where you're going. If you're about to enter a (US?) master's program, then I'd recommend reaching out to your potential advisor(s) for advice. Grad school is (or at least should be) vastly different from your undergrad experience. In undergrad, you fulfill requirements and scramble for grades, but a master's program should be more like a professional apprenticeship. Start forging relationships from day one, if not earlier, so when it comes time to write a thesis or study for exams, you're not introducing yourself to professors for the very first time. I think that reaching out to one or two with a paper you're thinking about turning into an article is a great way to introduce yourself.

Second, if you've only worked for a semester or two on this paper, chances are that it does in fact need some major overhauls. In grad school, you can seek out opportunities to present and workshop, which will help refine and strengthen your thesis. You'll also read a ton, typically a book per week per class. That should strengthen you're ability to identify and research relevant sources, interpret them, and situate them in an area of lively debate. I'm not being negative here—you're paper might already be great—but it will only get stronger and sharper with time.

Third, there's no rush, so you should be strategic. If you want to be in and out of academia, then think about making this the focus of your master's work. If you're thinking about an eventual PhD, then I'd recommend using it as a basis for a few conference presentations and maybe have it drafted into a manuscript for journal submission by December '19. That'll generate several CV lines and give you a really strong writing sample for your PhD applications.

At present, then, I'd recommend doing a couple things. First, go back over your paper and make any corrections that have been recommended to you and fix anything else that jumps at you. Then set it aside: after a few weeks, you'll be able to read it with a whole new set of eyes. Second, reach out to a few professors that you're hoping to work with. Let them know that you've got an undergrad paper that you're hoping to spin into a publication, and see what they recommend. Two professional-sounding questions might be: what kinds of workshop opportunities are there in the department, or do grad students mostly work one-on-one with professors? and, do they recommend reading any historiography before coming into the program, either for this particular paper or more generally?

That's probably where I'd focus my efforts at present: reading closely through a couple of the major works, and spending the time to digest and enjoy them thoroughly. If you still have university access, maybe surf through some major journals like the William and Mary Quarterly and the American Historical Review to see what's getting people excited and what other journals show up in their footnotes. History will soon seem more and more like a job—an exciting and rewarding job, I hope, but a job nonetheless. So take the summer months to relax and enrich your love of good historical writing, and that should pay dividends in the fall.