By ultimately offering a native language web experience to end users, the localization process -- understood as the adaptation of an existing website from a linguistic, cultural and technical perspective to render it multilingual, unavoidably contributes to the Web for All paradigm. However, to date, there has been little discussion about how and to what extent the localization industry is adhering to web accessibility (WA) best practices as part of their regular workflows to fully pursue that goal. This paper gives an account of the latter by reporting on the qualitative data gathered from a series of semi-structured interviews with 15 representatives of six different world-renowned language service providers (LSPs). Findings reveal that, while LSPs deal with web content and technology on a daily basis and now offer a broad spectrum of web-related services, including Digital Marketing and User Experience Design, conforming to WA requirements is not critical yet for the localization industry. We therefore explore why localization companies do not see themselves as key stakeholders in the value chain for web accessibility and we review which would be the main drivers for them to consider compliance with WA guidelines in the future.