You can run the Chrome App on a Chromebook which allows programming in Blockly and can be switched to creating Basic code.You can do the same through the PICAXE Cloud using Chrome or other browsers.
The second of the cycling advertising images for the Chrome App shows Basic editing -
https://chrome.google.com/webstore/detail/picaxe-blockly/hhdlapnjifkkcpghcapopejopnbpapnb
Basic programming using the Chrome App is not as full featured as using PE6. The best way to see if it suits your needs is perhaps to download it and try it.