Buisness Analyst Online (BAO) is now part of ArcGIS Online (AGOL). If you want to use BAO you will first need an AGOL account. See the UW AGOL page for details.
Once you have created your account, let the site administrator know that you want to be added to the BAO group. Make sure your message includes whether you are using the main UW AGOL subscription or the UW Supplemental AGOL subscription.
By default, users are assigned only 10 credits. Allocation of additional credits will be done on a case-by-case basis. In order to get more credits, you will need to estimate your usage and ask for the number of credits you expect to use. See AGOL credits to estimate the amount of credits you expect to use. Once you have made your estimate, send a message to the the site administrator with details.
IMPORTANT NOTE FOR GEOCODING: If you intend to do a large geocoding run, use Business Analyst Desktop instead, unless you have the cash to buy lots of AGOL credits.