I think the information about JLPT level is from Jonathan Waller‘s JLPT Resources page; I don't know if there's a way to search Jisho beta for all words that are tagged with a specific JLPT level, but you can for example go to: http://www.tanos.co.uk/jlpt/jlpt3/vocab/ to get a list of all the words. I don't have a wanikani subscription but if you do I think you can just go a page like this: https://www.wanikani.com/level/2 and it'd show you all the kanji and vocab from that level.
I know that's not exactly what you asked for, but maybe it helps.
This is possible! Like so: http://beta.jisho.org/search/%23words%20%23jlpt-n3
And for Wanikani: http://beta.jisho.org/search/%23words%20%23wanikani2
Yes. For example:
黙 #wanikani*
This gives you all wanikani words for 黙
黙 #wanikani46
This gives you all wanikani 46 words for 黙
黙 #jlpt
This gives you all JLPT words for 黙
黙 #jlpt-n3
This gives you all JLPT N3 words for 黙
黙 #common
This gives you all common words for 黙
黙 もく
This gives you all words for 黙 with the もく reading
Plus combinations of the above can be used.
Sorting by Tags
I notice the inclusion of things like "JLPT" level and "Wanikani" level. Since that would mean many items have been categorized as such, is it possible to see listings of say JLPT N3 kanji and such?