I live in Santiago, Dominican Republic and I can testify that indeed you ARE now required to present proof vaccination against the Coronavirus prior to entering shops, grocery stores, shopping centers, and fitness clubs.
This is happening to me right now. I just show them a picture of my vaccination card that I have saved on my phone. In one place (Plaza Internacional) I was also required to show a picture of my passport at the same time. Again, you will get denied entry to LARGE stores and shopping malls here in the Dominican Republic if you can not show proof that you have received the vaccine for corona virus (COVID-19). In the small shops they do not ask for it. And neither do they require proof of vaccination in small restaurants.