This is unlikely to pose a problem: most consent-requiring stuff happens only per JS, and consent must be opt-in anyway. However, accessibility concerns might be legitimate.
This brings us to the next reason: consent is opt-in. The ePrivacy directive requires consent for uses of cookies that are not strictly necessary, and GDPR Art 4(11) and Art 7 defines consent to be an opt-in through an affirmative action or expression that unambiguously indicates the visitor's wishes. Pre-checked boxes or “by continuing to use this page …” style consent mechanisms are uncompliant. If a website's mechanism for asking for consent doesn't work, it cannot assume that consent was given and is not allowed to perform the relevant data processing.
Does GDPR require accessibility? Not explicitly. Accessibility requirements can be derived from other laws. But the GDPR has some basic principles that point in this direction. As a special case for written declarations that also consider other matters, the request for consent must be in “an intelligible and easily accessible form” (Art 7(2)) which arguably also applies to textual media such as websites. More generally, the transparency principle in Art 5(1)(a) could be argued to imply accessibility for information regarding the data processing.
For the context of the data subject rights, Art 12(1) provides details on how information must be given (transparent, intelligible, easily accessible, clear and plain language) and Art 12(2) requires data controllers to “facilitate the exercise of data subject rights”. This facilitation means that data subject shouldn't face unreasonable barriers to the exercise of their rights. A thoroughly inaccessible website could be such a barrier, e.g. putting contact info into an image.
However, it doesn't follow that specific features such as keyboard accessibility or ARIA markup are required by the GDPR. Even if we interpret the GDPR to require some level of accessibility, a data protection authority or court would have to look at the context: what level of accessibility can reasonably be required?
This answer primarily covers ePrivacy/GDPR as I have no deeper understanding of the CCPA. However, CCPA is arguably more accessible, because it mandates specific technologies (links) and titles (Do Not Sell My Personal Information) which happens to be quite accessible. CCPA also authorizes and demands further regulations to ensure that notices and information are accessible to consumers with disabilities.