Does the radio frame have to be grounded? Well, no, but it won't hurt
Does the antenna? Depends.. Some antennas yes, they do need a ground, Some no
Even among Truckers CB is not as popular as it once was.. I'm 3 miles from a truck stop and interstate and am often surprised when the CB talks to me.
It is possible you have a problem.. How to figure out if you do (NOTE SOME Danger here)
If your Radio has a SWR meter function select FORWARD. then transmit, adjust the knob so you see a FULL SCALE reading, do this quickly
Now switch to REVERSE and transmit again note the meter reading and get back to me... Any reading over 2:1 indicates an issue, though the radio should work up to 3:1.
True story:
Motorola shop ordered a box of jumper cables,,, Pre-made at the Motorola factory.. EVERY cable, both ends, the connectors were shorted. EVERY ONE.
I had the same problem with my Towed car.. the umbilical one end was shorted.