Can civilians with Dbids on base be able to shop at the commissary or base exchange?

No.

The only people that can shop at the exchange or commissary are authorized patrons.  A DBIDS card is just a base access card.  Just because you got a DBIDS card to delivery pizzas to base housing doesn't mean you get to buy cheap food in the commissary.  If you want to do that, enlist or marry a servicemember.  

.Yes, if accompanied with an active duty Sailor. Check out the Navy Exchange anywhere in the World. I bought a new telescope, Subic Bay Naval Station Navy Exchange. I proved that Sirius ain't Jupiter to the officers and gentlemen of USS Somers DDG-34

A Dbid just gets you on base/post to do your job nothing more than that. 

There was a time period when they could go to the exchange during COVID if essential workers. But unless you qualify as a retiree, dependent or a veteran you cannot. 

Overseas you might be able to a exchange but it is limited. 

https://www.military.com/daily-news/2021/01/13/dod...

For the dude who said yes, that base shut down in 1992. It might have been an option then but not how.