Some web browsers are more accessible than others for users with disabilities. In order to be accessible, a web browser, like any other software product, must have an interface that is accessible. For example, the browser must be operable by keyboard only (for people who are unable to use the mouse), and it must be compatible with assistive technologies such as screen readers. In addition to having an accessible interface, web browsers must be capable of presenting web content in an accessible way, provided that the web author practiced accessible web design techniques. These requirements are explained in detail in the W3C's User Agent Accessibility Guidelines (UAAG) 2.0. UAAG, and its relationship to the Web Content Accessibility Guidelines (WCAG), are explained in the Knowledge Base article Are there accessibility guidelines for web browsers?.
Regarding the need for browsers to be able to present web content in an accessible way, one way that browsers can do this is by accurately communicating details about user interface components with assistive technologies. The website HTML5 Accessibility documents how well each of the major browsers performs in this area. However, communication is a two-way street. Just because a browser does well at exposing information to assistive technologies does not in and of itself guarantee accessibility. Assistive technologies must do their part to handle the information provided by browsers.
Particular assistive technologies work best with particular browsers. This is a moving target, but one source of information that can help identify trends in browser usage among screen reader users is the WebAIM Screen Reader User Survey. For example, Survey #6, conducted in July 2015, found that Internet Explorer (various versions) and Firefox were the most popular browsers among screen reader users.
For additional information, see the Knowledge Base article How can I tell whether a software application is accessible?