The nodes have built in support to send meshtastic messages on to the internet via MQTT, and that bridge is bidirectional. You could send a message like "GET / HTTP/1.1\r\nHost: ddg.gg\r\n\r\n" and have an MQTT listener fetch it and give you the reply.

It's go at a few KB/s with significant latency, and I think each packet is <300 bytes (as compared to Ethernet's 1500 bytes). So I doubt you'd enjoy that, especially with modern websites. Similar things could be done with other protocols.

I'm not sure if someone has built any of this or how reliable it is. Probably, because why not, right? But I haven't come across any such projects.