Decided by the US Supreme Court in 2002, this case  held that even requests for accommodation that might seem reasonable on their face, e.g., a transfer to a different position, can be rendered unreasonable because it would require a violation of the company's seniority system. While the court held that, in general, a violation of a seniority system renders an otherwise reasonable accommodation unreasonable, a plaintiff can present evidence that, despite the seniority system, the accommodation is reasonable in the specific case at hand, e.g., the plaintiff could offer evidence that the seniority system is so often disregarded that another exception wouldn't make a difference.
If you do get sued, if you immediately remediate your website, you may be able to get the lawsuit dismissed on mootness (there’s no longer anything in dispute, i.e. plaintiffs are arguing your website is inaccessible but you’ve already made it accessible). This definitely does not mean you should wait to fix your website but it does mean you may have an out.
Once you have been hired and started work, your employer cannot require that you take a medical examination or ask questions about your disability unless they are related to your job and necessary for the conduct of your employer's business. Your employer may conduct voluntary medical examinations that are part of an employee health program, and may provide medical information required by State workers' compensation laws to the agencies that administer such laws.