This page has Lichess puzzles with something similar to the Puzzle Storm mode, except you choose a fixed rating band to work on, and there are no time penalties or bonuses. It was cobbled together with chessboard.js (modified by me for my preferred mobile behaviour) and chess.js. Sounds and pieces are from Lichess.
This site does not store any of your results, or send them to Lichess. Once you select a rating band, a batch of between 5000 and 10,000 puzzles will be loaded, and presented in random order once you start the storm. Bands between 1000 and 1699 have the most puzzles, with 10 or 11 batches each, one of which is chosen at random to download. In the results table(s) displayed after the end of a storm, links go to the Lichess page for that puzzle, and clicking on one will open it in a new tab. The rating of a puzzle on Lichess may have changed since the database I downloaded was created.