For example, if I set this tool for English, then type /ʌ/ in the search box, it should return a list of English words that contain that sound.
This is basically a programming question, but there are very many such tools. What you primarily need is a phonetically-notated dictionary file, so first you have to decide on a particular phonetic alphabet (the letter "ʌ" is ubiquitous). You can use the CMU dictionary. If you specifically want IPA, you need an interface that translates user-input IPA letters to the correct CMU notation. In the case of "ʌ", that is the primary-stressed allophone of a vowel including "ə", so you would search for all instances of " AH1 ". There are many ways to do that: one that doesn't involve writing code is to use Jedit and do a regex hyper-search, which will return some 7,000 examples.
You can also try this tool, which may do what you want, which will lead you to various dictionaries (there isn't just one "English").
Geoff Lindsey (a British Phonetician) maintains CUBE, a pronunciation dictionary of Standard Southern British English, and his YouTube channel has a video explaining its use, with instructions on how to search for specific pronunciations starting around 6:55 (note this search field only allows standard ASCII characters as input, but uses an idiosyncratic mapping, rather than e.g. X-SAMPA). In the case of words containing /ʌ/ you would search for "y".
Note that words here are by default transcribed according to his transcription of contemporary SSBE, although if you tick "trad" under systems it will also give the more traditional Received Pronunciation.