Verdict: Yes
Yes, through your browser. Lucidchart is browser-based diagramming software used at lucid.app, so it works on Linux with no installation at all.
Lucidchart was built as a web app. Lucid's own product page points you to lucid.app in your browser and describes real-time collaboration on diagrams there. That is the whole product, not a cut-down version of something you install.
Because of that, moving to Linux changes nothing. Open Firefox or Chrome, sign in, and your flowcharts, org charts and network diagrams are as you left them. There is a free tier if you only make the occasional diagram, as well as the paid plans.
Your existing diagrams stay on Lucid's servers, so there is no file to move. If you want copies for safekeeping before you wipe Windows, export the ones you care about as PDF or PNG, which open on Linux with the viewer that comes with the desktop.
If you would rather have a diagram tool installed on your own machine, LibreOffice Draw handles flowcharts and simple plans at no cost, but it works alone rather than with other people at the same time.
Free, installed on your computer and fine for flowcharts and diagrams. No live collaboration and the shape libraries are plainer.
Your files: Cannot open Lucidchart documents. Opens PDF and image exports, and can import Visio .vsdx files.
Browser-based and free, aimed at design and interface work. It can serve for diagrams but it is not built for them.
Your files: No Lucidchart import. Bring diagrams in as exported images.
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