I have a 12 V regulated power supply that I use for a car audio amplifier in my home and I only have it hooked up with a cord that has a line and a neutral, no ground. Could this be dangerous or cause any other issues? It's been working fine so far and I don't get a shock when I touch the case of the power supply.
Quite simply, they provide an earth connection so it requires connecting to earth.
The safety earth is there to protect you, and also the equipment, in the event of a fault that connects a dangerous voltage to the chassis and metalwork.
The fact that no such fault has happened so far does not stop it happening in the future. On the day it happens, if at all, you'll need the safety earth. The consequences of not having it range from pain to serious harm to death.
Take no pointless or easily-avoidable chances. Use a proper mains cable with earth and change the risk from unlikely or improbable to nigh-on impossible.
It's a good lifelong habit to adopt and will cost you all of about £2 for a cable.
Some appliances may give a shock if there's a wiring fault. Some, will only give a shock if there's TWO wiring faults. Each may be safe if there are no faults.
But, local safety standards (which differ according to century and region) may require... almost ANYTHING.
If you might touch the 12V output wiring of the power supply, and if there is any hazardous AC voltage involved, a ground-fault interrupter (GFI, or GFCI) and the two-prong plug should make the system shock-safe. Any modern three-prong power plug has a ground that would also make the system shock-safe (if the design passes modern safety standards that I'm familiar with).
There is no absolute safety, so there's no possible complete answer to this question. Notably, even if there's NO AC power involved, batteries have been known to catch fire.