It depends on what you mean by "self-taught" and "mathematician".
What is definitely true is that you probably have the chops to learn any particular branch of math that you are interested in. And seeing as you enjoy math, I'd say you should certainly keep on learning it. So if by "mathematician" you simply mean "one who studies and does math", then the answer to your question is "Yes, you can be a self-taught mathematician."
If, however, by "mathematician" you mean "one who does original research in mathematics" or "one who does mathematics professionally," then the answer is considerably less affirmative. Mathematics is nowadays a vast field, and getting to the frontier is a formidable challenge. Almost all the material you find in textbooks or on Wikipedia are many decades (or centuries) old, and so even if you master all the textbooks in sight, you will still need to wade through the contemporary literature to find out what is going on nowadays. Moreover, there is an art to doing research in knowing how to ask an interesting question, which is something that can't generally be well learned from any reference.
The way that most mathematicians learn how to find a frontier of mathematics and push beyond it by asking questions is, simply, to follow in the footsteps of their Ph.D. advisor. If you are serious about being a mathematician in the more committed sense, you will almost certainly need to still get a Ph.D. So then the question is "can I teach myself enough mathematics to get into a Ph.D. program?" The answer here is again "yes". There are certain things that you are expected to know going into a Ph.D., but they are all quite attainable. Given your stated background, I'd say the things you should learn to have a background equivalent to a undergraduate degree in mathematics are:
- Real analysis, at the level of Rudin's "Principles of Mathematical Analysis" or Strichartz' "The Way of Analysis".
- Abstract algebra (groups, rings, and fields, at least), at the level of maybe Grove's "Algebra".
- Topology, at the level of Munkres' "Topology".
- A little complex analysis and differential geometry would also be good, but probably not strictly necessary.
If you're definitely not interested in getting a Ph.D., but still want to try your hand at research in mathematics, there are some things you can do... If you want to hear more about this avenue ask me in the comments to elaborate.
EDIT: To elaborate on the "no Ph.D. route," I can think of several difficulties that you will have to overcome in trying to do research-level mathematics:
- Be reasonable
First of all, bear in mind that to start doing research level mathematics from your current state, you will need several years of dedicated reading and working out hard problems. Above and beyond enthusiasm, it will also take a lot of work.
- Finding the frontier.
I alluded to this already above. If you'll permit me a colorful metaphor, I like to think of mathematics as some foreign continent that is only partially charted. Textbooks and encyclopedias provide maps of the territory, but there are large blank areas labelled "monsters" at the fringes of what's known. Beyond that point is the domain of the research mathematician "cartographers". Finding your way around those blank areas is tricky because much of it has already been explored, but it's not on the maps--the knowledge is in the pockets and scribbled on the personal charts of the mathematicians who explore it. If you stake out on your own without consulting the experts who went before you, you'll likely find yourself trudging through the same tedious jungle that others trudged through decades before, but without the benefit of being able to claim priority in exploring it first. It's annoying to get to the summit of some mountain and find a flag already planted there from the 70's.
So be judicious in what you set your mind to. Talk to people and read the literature to get a feel for a field before you start trying to make contributions. Also, be aware that some fields are much easier to get into than others. Combinatorics, for example, has a reputation for having a low entrance threshold. New fields and fields without as many practitioners are also generally easier to get to frontier of (e.g. integer-point enumeration in polyhedra is easier to get into than analytic number theory).
- Get feedback as you go.
This is very important: If you just read books and do math by yourself, without talking to other mathematicians, you are liable to develop bad habits, repeat the same errors over and over again, and fail to develop the necessary communication skills to share your work in a useful way. In other words, if you don't talk to anyone as you go, you are liable to become a crank.
Sites like math.stackexchange, and mathoverflow even more so, are good tools for interacting with other mathematicians. I highly recommend that you start trying to solve problems on MSE. You should also start reading questions and answers on mathoverflow to get a feel for what's happening in research these days, but don't even bother trying to contribute there until you've got a few more years of reading under your belt. Beyond MSE and MO, though, you should also find mathematician friends that you can just talk to and bounce ideas off of. You might be able to make such friends through these sites, but I can't speak from personal experience on this matter.
I also recommend taking/auditing classes whenever you get the chance as you're trying to learn more math. The feedback loop is faster for classes than for most other communication channels.
- Don't quit your day job.
No one is going to pay you just to do math if you don't have a Ph.D. (Even most Ph.D.'s don't get paid to just "do math.") If you want to make any sort of career out of this, you'll have to find a useful application. There are lots of "applied math" jobs to choose from. These can also be directly useful to your research aspirations, as often times you naturally bump into interesting math questions in the course of investigating some application. (C.f. Claude Leibovici's answer.)
Some further random thoughts:
- Don't try to prove the Riemann Hypothesis. Don't try to prove Fermat's Last Theorem. Don't try to prove any famous open problem. Until you establish yourself as a productive research mathematician, even mentioning these names under your breath will automatically put you in the "crank" category (Crk).
- If and when you get to the point of writing papers, ask somebody for help in deciding where to submit. Picking a journal is an art unto itself that is not worth mastering. Alternatively, you can not worry about submitting at all, and just post to the arxiv.
- Don't post to vixra. Totally useless, and another automatic "crank".
- As long as your job doesn't depend on it, don't obsess over publications. There are other ways to make productive contributions to the corpus of mathematical knowledge (e.g. MSE, or just talking to other mathematicians).