In my kitchen I have an under counter fold up TV, but it only has a cable connection to it. Is there anything out there that works with just a cable connection?
If you mean an RF/Coaxial cable connection, no - there is no direct support.
You might want to verify that it only has an RF/Coaxial input.
You could get an HDMI>Coaxial RF adapter for $40:
If it has Composite (RCA - red/white/yellow) connectors you could get a cheap $8-15 HDMI>Composite converter and use it with any model:
I'd heavily recommend replacing the TV with a small modern flatscreen with HDMI inputs.