Twitter friend @Ebradley127 pointed out Coffeetivity, a site that gives listeners multiple coffee shop soundscapes to choose from. It offers such aural locations such as "morning murmer," "Lunchtime Lounge" (slightly more upbeat and chatty), and "University Undertones" (which i'm guessing includes the sounds of elbow patches on leather chairs). There are also 'premium options' like "Paris Paradise," "Brazil Bistro," and "Texas Teahouse." Those options are locked away at a cost of $9 for a year's worth of additional bustling cafe sounds.
I am more partial to the website Rainy Cafe. There is nothing to buy or upgrade, just two options: rain and cafe sounds. The rain storm includes some rolling thunder, so it may be a little ominous or stress inducing for some. The cafe tonally feels more subdued that Coffitivity's versions, and it seems to have slightly more bass with slightly less dish clattering sound.
If you want to add a little extra, you can also load up Coding.fm. This site features looping streams of typing on keyboards. It may seem a little inane, but it can be a nice added boost- and in my experience brings those "cafe sounds" to true life as there are usually armies of people on their laptops there anyway. Nothing fancy, not even separate volume controls. Just 3 variations on typing.
One final trick that I found enjoyable was to pull up some instrumental guitar or piano music to add the effect of a live performer. Actual live music in a restaurant sets me on edge, but when you know you're getting performance, it can be quite entertaining without being distracting.
I'd recommend trying it out with Will Ackerman's "In Search of the Turtle's Navel." It's an instrumental acoustic guitar album from 1976 which launched the New Age record label, Windham Hill. Simple but enjoyable. Just cue it up on your streaming service of choice and set the volume fairly low to balance with the other sounds. Easy.