The most straightforward path is through their own website, but they keep a pretty tight paywall after a few free articles each month. I tend to hit that limit quickly because the essays are so long and dense—you can't just skim one.
My local library subscribes to the digital magazine service PressReader, which gives me full, legal access. It's a bit clunky to navigate on a browser compared to a proper news app, but having the complete, high-quality PDF layouts feels closer to reading the physical magazine. For deep archival stuff, like reading a seminal review from the 1970s, I've had decent luck with JSTOR if you have academic access; they have a huge run of it, though the interface is built for citation hunting, not casual browsing.
Honestly, the 'archive' part of your question is key. The NYRB's own online archive only goes back to 1993, which misses decades of its most famous pieces. For anything earlier, you're looking at those third-party database services or hunting in second-hand bookshops for their anthologies.