What web application is most suitable for learning the common non-English visual language used by the educated U.S. Deaf community?
As a native sign language user I highly suggest you start with lifeprint.com it is excellent for the free YouTube videos and the dictionary explains the many nuances and uses of each word. It helps a lot to see the signer from two angles, and having a newish signer in the videos helps people following the videos learn about and avoid the most common mistakes.
There is also signingsavvy.com which is subscription based, helpful for people without English background or visual learners since there are images associated with each word. It has an app as well, but no images on the app version unfortunately.
Once you've learned enough, you can start watching videos posted to the ASLThat FB page (no subtitles or voice audio, you need to know the basics of ASL to follow it) and there's also several great communities on Reddit: r/asl & you can get someone to practice with online at either r/deafskype or r/aslpenpal.
howyousign.com is a great launchpad to get you started on discovering different resources available for learning ASL or the interpreting profession.
Last but not least, seek out Deaf Night Out events in your location and try to get a real life deaf buddy who's willing to mentor you. DNO tends to happen on Friday nights at either a cafe or a pub. It's the best way to learn. Books (even current ones) will teach outdated signs, or rather signs that are not used regionally. Sign language varies a lot, by country, and even within a country there are many accents or dialects just like spoken languages. For example, there are more than 21 different signs for the word Strawberry.