Here's a list of stores we've confirmed are still open (closings and hours are subject to change):

Albertsons: Some stores have adjusted hours, depending on location. Every store is open from 7 to 9 .am. for the elderly and vulnerable to shop.
Aldi: Stores are open from 9 a.m. to 7 p.m.
Big Y: Stores remain open from 7:30 a.m. to 8 p.m.
BJ's: Stores are operating with normal hours, however shoppers should "expect some shipping delays on minor portions of summer seasonal items.
Costco: Stores are operating with normal hours.
Cumberland Farms: Stores are operating with normal hours.
CVS: Stores and pharmacies are operating with normal hours.
Dollar General: All stores will close one hour early. The first hour of operation is reserved for senior citizens.
Food Town: Stores are open from 8 a.m. to 8 p.m. Anyone 65 and older can shop from 7 to 8 a.m. to avoid crowds.
Fresh Market: Stores are operating with normal hours. All seniors and at-risk customers can shop 8 to 9 a.m.
Gelson's: Stores are operating with normal hours. Anyone 65 and older can shop from 7 to 8 a.m. to avoid crowds.
Harris Teeter: All stores close at 9 p.m.
HEB: Stores remain open from 8 a.m. to 8 p.m.
The Home Depot: All stores close at 6 p.m. In-store classes and workshops are canceled.
Kroger: Adjusted store hours vary by location, but most are open from 7 a.m. to 9 or 10 p.m.
Lowe's: Stores are operating with normal hours.
Publix: All stores close at 8 p.m.
Ralphs: All stores are open 8 a.m to 8 p.m.
Safeway: Some stores have adjusted hours, depending on location. Every store is open from 7 to 9 .am. for the elderly and vulnerable to shop.
Sam's Club: Starting Tuesday, March 17, stores will be open 9 a.m. to 8 p.m. Sunday hours will remain 10 a.m. to 6 p.m.
Shoprite: Stores are operating with normal hours.
Stop & Shop: All stores open 7:30 a.m. to 8 p.m. Online pickup has been suspended. Anyone 60 and older can shop from 6 to 7:30 a.m. daily.
Target: All stores close by 9 p.m. Every store is dedicating an hour each week for elderly to shop.
Trader Joe's: All stores are open 9 a.m. to 7 p.m. Select stores will open early for elderly and at-risk customers.
Walgreens: Stores and pharmacies are operating with normal hours.
Walmart: All 24-hour stores have reduced hours, opening from 6 a.m. to 11 p.m. Stores currently operating under more reduced hours will keep their normal hours.
Wegmans: Some stores have adjusted hours for overnight cleaning, depending on location.
Whole Foods Market: Some stores have adjusted hours, depending on location. All stores open one hour early for elderly and at-risk customers to shop.
Winn-Dixie: All stores close at 9 p.m.

Here's a list of stores that have closed for the time being:

Abercrombie & Fitch
Apple
Anthropologie
Bloomingdale's
Everlane
Free People
Glossier
J Crew
Lululemon
Lush Cosmetics
Macy's
Madewell
Nike
Nordstrom
Patagonia (Note: They have also suspended their online shopping services.)
REI
Urban Outfitters

What about delivery options?

Many stores, including Stop & Shop and Safeway, are still fulfilling online orders for delivery. Due to high demand, stores warn that there may be a delay in deliveries — and there's even the possibility that these services will be suspended if the COVID-19 outbreak worsens. As a general rule, call your local store first before you place an order.

Walgreens and CVS, however, are making it even easier for customers to get their prescriptions filled. Both pharmacies are waiving all delivery fees, making it possible for any elderly or at-risk people to receive their medications without leaving their home.

(source: Good Housekeeping)