Any VPN that terminates on the firewall (be it site to site or remote access / “road warrior”) may be affected, but not all will. Some VPN tech uses very efficient computations. Notably affected VPNs are OpenVPN and IPSec / StrongSwan.
If the VPN doesn’t terminate on the firewall, you’re in the clear. So even if your work provided an OpenVPN client to you that’s affected by AES-NI, because the tunnel runs between your work laptop and the work server, the firewall is not part of the encryption pipeline.
Another affected technology may be some (reverse) proxies and web servers. This would be software running on the firewall like haproxy, nginx, squid. See serverfault.com/a/729735 for one example. In this variation of the check, you’d be running one of these bits of software on the firewall itself and either exposing an internal service (such as Nextcloud) to the internet, or in the case of squid doing some HTTP/S filtering for a tightly locked down network. However, if you just port forwarded 443/TCP to your nextcloud server (as an example), your nextcloud server would be the one caring about the AES-NI decrypt/encrypt. Like VPN, it matters to the extent of where the AES decrypt/encrypt occurred.
Personally, I’d recommend you get AES-NI if you can. It makes running a personal VPN easier down the road if you think you might want to go that route. But if you know for sure you won’t need any of the tech I mentioned (including https web proxy on the firewall), you won’t miss it if it’s not there.
Edit: I don’t know what processors you’re looking at that are missing AES-NI, but I think you have to go to some really really old tech on x86 to be missing it. Those (especially if they’re AMD FX / Opteron from the Bulldozer/Piledriver era) may have other performance concerns. Specifically for those old AMD processors (Not Ryzen/Epyc), just hard pass if you need something that runs slightly fast. They’re just too inefficient.