Is there a way to connect a DVD/VCR combo unit that does not have a built-in tuner to a TV that has cable but doesn't have a cable box so that it can record TV programs?

I've never heard of a VCR without a tuner, unless it is just a "player" and not a recorder.

How would you record anything without a tuner?

If what you have is a player only, you can't record with it.

I've never heard of a VCR without a tuner, unless it is just a "player" and not a recorder.

How would you record anything without a tuner?

If what you have is a player only, you can't record with it.

I also do not have a cable box but do have a direct cable hookup.
With my previous VCR/DVD combo unit I was able to record from my TV to the VCR and also play VCR tapes and DVDs. This was hooked up with coaxial cables and a mini modulator. I bought a new unit and a new modulator however all I can do is play tapes and DVDs. I mentioned several times to the salesperson at the store that I did not have a cable box but apparently it fell on deaf ears. It wasn't until I hooked everything up that I discovered I could not record. When I called the store I was told by the salesperson I couldn't record if I didn't have a cable box. If that's the case, why was I able to do it with my previous unit. My TV only has an antenna in jack. Can you explain?

Your other unit probably had a tuner built into it.

Without a tuner, you cannot record. The unit can be capable of recording from a video connection (to answer the older question above).

With basic cable or antenna, you need a recorder with a built-in tuner. Or, you need a separate tuner box... like a cable box. This makes things more difficult for setting up recordings however.

The modulator you picked up is to convert the video connections that your DVD/VHS combo unit uses, to the RF (or coax) connection that your TV uses.