It should be possible to reach the Developer App by going to https://developers.facebook.com/apps or https://facebook.com/developer.
If this does not work, first ensure that you are logged in to your personal Facebook account. Facebook does not allow creation of apps from a business account. Once you create the app you will be able to add other people as app administrators, developers, or testers. It is also possible to convert the business account to a personal account if you don't already have one. Note that a Facebook business account is not a Facebook account for a business; it is an account for an individual that is restricted to only certain business activities.
What is the difference between a business account and a personal account?
Business accounts are designed for individuals who only want to use the site to administer Pages and their ad campaigns. For this reason, business accounts do not have the same functionality as personal accounts. Business accounts have limited access to information on the site. An individual with a business account can view all the Pages and Social Ads that they have created, however they will not be able to view the profiles (timelines) of users on the site or other content on the site that does not live on the Pages they administer. In addition, business accounts cannot be found in search, send or receive friend requests, or create/develop apps.
Next, ensure that you have verified your account with a mobile phone or credit card. If your account is not verified, or you have only verified your account using another method, then you may not be permitted to add any new apps.
How do I verify my developer account?
Facebook has several features in place to limit the potential for spam and abuse on the site. Verifying your Facebook account helps you retain control of your account and apps on Facebook. There are two ways to verify your account. First, you can verify your cell phone, which allows us to verify that a real person is in control of the account. This process is explained here.
The second way to verify your account is to add a credit card to your account. You may do so here.
Unfortunately, if you cannot verify with a cell phone or add a credit card, you may be unable to verify your account. All developers must be verified before they can create a new app.
Also verify that you have not turned off Facebook Platform on your account. From Privacy Settings go to Apps, Games, and Websites. If you are using any third party apps or have the option to turn off all apps then Platform is enabled. You might also verify that you have not blocked the Developer app: from Privacy Settings go to Manage Blocking and look under Blocked apps.
You can also check the Developer app profile page which should list some options on the left. If it thinks that you have already added the app then there should be a Remove App link.
If Facebook did allow two accounts, your second account would have to be another personal account if you wanted to use it to develop an app (a clear violation of Facebook terms, since a personal account always includes a personal profile). Technically, I suspect that it has to be a personal account so that it it will have a profile to use for issuing access tokens, which are needed to use apps (such as the Developer app, and your own app that you are developing). So to avoid violating the terms you will need to use your existing personal account to develop an app. Although a personal account is required to develop an app, a page can be administered using a business account if there is no need to use the app with that account.
Note that a business page and business account are not the same; you may create as many business pages as you like using a single account (login/password). Unless you set yourself as a "Featured Page Owner", the page will not link back to your profile, and your profile will not link to the page if you don't "like" or share it. As a page admin you can also post and comment as the page, even though you are logged in to your personal account, and these posts and comments will link to the page and will not appear on your personal profile or in the News Feed of your friends (unless they "like" the page). So most people will see no association between the page and your personal profile. Only other page admins will be able to see you in the list of page admins. You can add as many other admins as you want, and after adding other admins you can remove yourself as a page admin if you want to completely sever all ties between you and the page.