I will take a slightly different approach, as I think you could argue that you already have started (but you just didn't have the "official title" of Business Analyst").

Have you ever looked at a process and asked if there was a better way?  Have you researched a problem and provided evidence based decisions to a manager or peer to influence a decision?  Have you listened to customer or stakeholder input and then worked to get a solution to their problem implemented?  If you said "yes" to any of these things, welcome to the world of being a business analyst!

The real challenge is getting hired into the position once you decide that it is a fit for your next career move.  I would suggest doing the research within the IIBA and also start searching within your company and professional network to see who else you might know that is a Business Analyst.  You can also check for "Product Managers" as well, as they often have similar responsibilities.  The other path that I would strongly suggest exploring is to get in touch with recruiters in your area that hire for contract positions.  There is a growing trend of bringing in BAs on contract for larger projects that have a finite term (6-24 months).  If you can find a few good recruiters they will be excellent resources to provide an honest assessment of your current resume, and gaps you could fill, and if all goes to plan could put you in front of hiring managers for a Business Analyst position of your very own!