According to TAMRON TAP-in Console (Model TAP-01) at the bottom of the page, the software is supported on Windows 7 through 10, and Mac OS X 10.10 (Yosemite) and 10.11 (El Capitan). (Note that the latest OS X is 10.15, Catalina, so the last Mac version being 10.11 is suspicious)
It may work under Wine, but I have no direct experience with Wine or this product. I just know that somethings do work under Wine and other things don't. But given that running Wine won't cost you extra $, it would be a good place to start.
It should work under VirtualBox because in that case you would be running a full version of Windows (that is legally licensed, right?!??!? If you need a license just for this, I would buy one from eBay. I have bought W7 licenses from there that MS is happy with) I have run various versions of Windows under VirtualBox and accessed external hardware with no issue.
At the top of the page, Tamron says
By attaching a TAMRON TAP-in Console and running the dedicated
software on your personal computer by USB connection, you can update
firmware in selected Tamron lenses and customize various lens
functions to meet your needs simply.
And from the image it looks like this is a piece of hardware that connects via USB to your computer. I can't guarantee that this hardware will work with VirtualBox, but I would guess that it's just a simple serial comms connection - in which case you should be OK.
Note that at the bottom of the same page is this:
Initial firmware update by Tamron service facility is required to
enable compatibility with the TAP-in Console.
I have no idea what this means, but it does sound ominous. The statement links to another page that basically tells you to go talk to your local Tamron distributor.
The Online Help has a lot of information, but I did see this important bit of information:
[CAUTION] Please note that you cannot use the TAP-in Console with any
lens other than those specified in the list of compatible lenses.
Which is found at List of Compatible Lenses (2019.6), which also says:
For Users of SP 35mm (Model F012) and SP 45mm (Model F013) Initial
firmware update by Tamron service facility is required to enable
compatibility with the TAP-in Console.
Finally, from what I just saw online you can buy this device for about $60. But what surprised me is that you can also rent it from LensRentals for $18/7days.