Just got a request for Visio, and as far as I can see there is a paid filter for Trados, while memoQ and Wordfast can do it.
Arguments for a Visio filter
Today I got may first job containing some Visio diagrams. So I'd like to ask too: Is Visio support planned for the future?
In the meantime, this is what I write in the File Formats reference document:
Microsoft Visio is a diagramming and vector graphics application that is part of the Microsoft Office family.
If you have Microsoft Visio, you can use TransTools For Visio that allows you to extract translatable text from Visio drawings (all formats supported by Microsoft Visio) and to merge translations back into the original Visio drawings.
CAT tools that allow you to handle Visio files via a round trip: Wordfast Pro 3, Okapi Framework, MemoQ, Déjà Vu.
To the above, I would add a new, and probably better option that has occured in 2020 (the reference files will get an update in the coming weeks):
XLIFF Manager: https://maxprograms.com/products/xliffmanager.html
OpenXLIFF Filters hanle Visio files, among many other formats.