My printer is on a vlan that has no internet access.

I have a Brother color laser printer (technically it's like LED or something? not home, don't have the exact model handy). It has built in wifi and ethernet for network printing. The wifi isn't configured, and the ethernet is manually configured with a static IP for my LAN... but no gateway address. This breaks outgoing network connections to the internet (as evidenced by the printer's inability to check for firmware updates), while behaving otherwise normally for all my LAN devices. I hope this info is useful!

I'm not sure of any specific printers but do be aware of the micro yellow dots because printers will do that and it gives away printer information such as the model and stuff.

Basic Brother black and white lasers come with only a USB connection and are really cheap to operate. Also no special drivers required.

I can't say anything in regard to modern printers, but I don't trust any proprietary device connected to the internet. I have an old inkjet and an old laser printer, both without network functionality themselves - but I connected them via USB to a Raspberry Pi, which runs a printer server and makes them available on the local network. I can print and scan from my Linux laptop, and I also managed to print something from my Android phone. Haven't tried Windows yet. Configuration of the Raspberry Pi was easy, especially enabling scanning via the network.