I think you would need to provide more detail to know what you have. Does it have a model number on it anywhere?

No problem! Mint XFCE sounds perfect to me.

That covers a pretty wide range of hardware, but that era would be around 2009-2015, give or take, so you would be looking at around Intel 1st gen to 6th gen most likely (Let’s be honest, there is nearly zero chance institutions would be using anything but Intel in that era). Pentium-branded CPUs from that time range, unfortunately, likely means low-end dual core CPUs with no hyperthreading, so 2C/2T, but I have run Linux on Core2-era machines without issue, so hopefully the CPU specs will be okay.

2-8GB of DDR3 RAM is most likely for that era, and as others point out, will be your biggest issue for running browsers. If the RAM is anything like the CPUs, I am assuming you will be looking at the lower end with 2-4GB depending on how old the oldest machines you have are, so I second the recommendation of maybe consolidating the RAM into fewer machines, or if you can get any kind of budget at all, DDR3 sticks on ebay are going to be dirt cheap. A quick look and I see bulk listings of 20x4GB sticks for $26.

In terms of distro/DE, I second anything with XFCE, but if you could bump them up to around 8GB RAM, then I think any DE would be feasible.

Hard drives shouldn’t be an issue I think, since desktop hard drives in the 320GB-1TB range would have been standard by then. Also, you are most likely outside of the “capacitor plague” era, so I would not expect motherboard issues, but you might open them up and dust them out so the fans aren’t struggling. Re-pasting the CPUs would also probably be a good idea, so maybe consider adding a couple $5 tubes of thermal paste to a possible budget. Polysynthetic thermal compounds which do not dry out easily would be preferable, and something like Arctic Silver 5 would also be an era-appropriate choice, lol.

I am not sure that I can really call what I did distrohopping, but

Mint w/ Cinnamon (several years ago on an old junker laptop and never ended up using it as a daily driver) -> Manjaro w/ KDE Plasma (daily driver for ~1 year) -> Arch w/ KDE Plasma (~2 years and counting).

I have also used Debian with no DE on a file server I made out of an old thin client PC and I have used Rasbian on a raspberry pi.

I had a similar issue. I actually wrote myself a text document listing out all of the programs I generally use post-install and any additional setup I did, so that way whenever I am setting up a new system I can quickly refer back to it and save myself a lot of time over doing one-off installs as I run into them.