EDIT: I apparently fell for a fake news Lemmy community. Its explained in the sidebar. I gave the OP the benefit of the doubt that they couldn't link the real new story, but trusted the attribution to APNEWS. If you enjoy participating in OPs social experiment on posters you're in the right place. I'm not, so I'm out.
The article said they don’t even have access to a toilet unless they are on the clock, so No, no showering facilities.
The context in the article where that line was taken was meaning: access to toilets after leaving the DC and accessing the new housing.
An employer isn't going to stop you from relieving yourself immediately after you clock out.
I'm not sure about your personal work experience, but for folks that work jobs that get you dirty many employers offer shower facilities on-site for employee use. The article says this housing pilot project is in a Walmart distribution center (warehouse). A quick google search shows that many Walmart DCs have employee shower facilities available. Here's evidence of that from an indeed poster:

Now, I don't know for sure that this DC where the pilot is taking place has showers. I also don't know if those using the housing are allowed to use the showers. However, the possibility exist, and I think Walmart would want its folks sleeping in company owned sleep pods to be bathed, so I think its highly likely that they chose a pilot location with showers, and I would guess they would allow the folks using these sleep pods to bath. It is not in Walmart's interest to have dirty/stinky workers showing up the next morning for work.