Can Time Machine roll back the operating system?

You can install the target version, then migrate your contents from Time Machine.

This migration from the Time Machine backup will be reliable (32-bit app discussions aside, see below) so long as the target OS X or macOS version is equal or newer than what’s in the Time Machine backup. Upgrades here work fine. Downgrades are not supported, not easy, and basically a whole lot of manual work. Downgrades are a hassle as apps and preferences can and do get modified after the upgrade to whatever OS X macOS version the backup was created with, and that doesn’t downgrade. You have to find and manually back those out, and start over there.

I’d try to get to upgraded to El Capitan OS X 10.11 minimally, preferably High Sierra macOS 10.13, and to Mojave macOS 10.14 in the best case. This for various reasons.

A Mac will not run a version of OS X or macOS earlier than the initial release of the Mac.

You will not get Yosemite or any other x86-64 version booting on a new Mac with Apple silicon.

The app will need to be 64 bit, for use on any Mac running Catalina macOS 10.15 or newer.

Mojave mdOS 10.14 is the end of the line for 32-bit apps.

If the Mac you buy is newer than the operating system you are trying to run on it, then you cannot run that OS on it. However, if you buy a mac that released before that OS released (in your case, before 2014), then you should be able to run Yosemite on it.

Macs, unlike iOS devices, do not prevent downgrades.

I suppose you could potentially also use a virtual machine to run Yosemite on a newer macOS version, but I don't know too much about that.

If you're not sure if the software will run on newer operating systems, then I'd install a newer OS on a secondary drive, then attempt to download and run the application. Just note that any macOS version newer than Mojave does not support 32 bit apps.