Can anybody recommend a cheap computer that can run Linux and has an Ethernet port with built in LTE or 3G?

First off, 3G is either dead or dying. All major U.S. carriers, for example, shut down 3G service in 2022. Globally, it’s probably similar. So your minimum is LTE.

To answer your question, no. Unless you happen upon a used device and the seller doesn’t know what they have, this is not cheap tech. And when it is cheap, it is often because it has vendor locks.

For example, I once bought a Datto DNA-VZ5 (a rebranded Axiomtek NA361) from a recycler. It had six Ethernet ports (4 x Intel i354 + 2 x Intel i210), Wi-Fi (Qualcomm Atheros QCA986x/988x, which is AC standard), and a cellular modem (Sequans VZ20M). The problem is, there are no open-source drivers for Sequans VZ20M. Further, the U.S. version is locked onto Verizon, and the Canadian version, onto Rogers. Further still, the device is somewhat larger than a typical desktop router, actively cooled, painted bright blue, and has the total of five antennas (three for Wi-Fi and two for cellular) on two opposing sides of the case. So even if you were able to make it work (for example, by replacing the Sequans card with a more open-source-friendly one), it still wouldn’t meet your aesthetic criteria.

Here’s what you can do. Find out what devices your cellular provider offers, then see if you can find those or similar in the secondary market… Most likely, you will end up with a standalone cellular modem. You may be able to find a cellular modem implemented as a PCI card (which you can install into any PC with a PCI slot; those are usually SFF or larger) or as an m.2 / mPCIe card (which you can use in a mini-PC if you take out the Wi-Fi card and replace Wi-Fi antennas with cellular ones). But either way, you’re highly unlikely to keep it under USD 100…