Answering mostly because I'm unhappy about being quoted out of context --
It has always been allowed, as far as I can tell since the day it opened in 1924, although passage across has been interrupted several times by wars, epidemics, etc.
Originally, the Causeway used to have a "proper" pedestrian sidewalk all the way across as well. But in 2008 Malaysia opened a new, hugely expanded Customs, Immigration and Quarantine complex located several km further away from the border, which did not have any provisions for pedestrians, and IIRC they even announced that pedestrians would not longer be allowed at some point. However, as far as I know this wasn't ever really enforced, and since the border remained congested as ever it remained a useful safety valve. It's a long, unpleasant and dangerous walk though, so my "can't really walk across" was referring to this being impractical especially if you have luggage etc, not legally banned.
During COVID, from early 2020 to 2022, the Causeway was closed to all travellers, vehicular or pedestrian, except a thin trickle of people with permits that were shuttled back and forth on special buses. With the lifting of COVID measures pedestrian access became possible again, and this is what the "since March 2022" references are talking about.