Are you listening to the built-in AM radio, or plugging one into the inverter?
If you are plugging one into the inverter, and if it does connect to the audio system, you should isolate that using an audio isolation transformer. Common mode chokes on the audio and/or power might work, too. If not, it's a power filtering problem. The inverter doesn't put out a clean sine wave like wall power. It approximates it with square waves (called "modified sine wave", which is OK for lightbulbs but bad for almost anything else. It takes a big filter to smooth out modified sine wave power. Last time I looked into it, I determined it wasn't really practical. If the radio has a transformer that puts out DC power, then it's a lot easier to filter DC power. You can use some capacitors and/or a common mode filter.
It could be that whatever you are plugging into the inverter is causing interference. Try an incandescent light and see if it's still a problem.
If the built-in inverter is causing interference with the built-in AM radio, then there's not much you can do except replace the inverter or the radio. It's possible that one or the other is defective, but I doubt it, and I would suggest getting a high quality standalone inverter instead. If you don't want to replace the inverter, I have a few suggestions:
Inverters operate at a high frequency internally to convert 12V power to high voltage, which they then chop up to make the modified sine output (probably similar for a pure sine inverter, too). It sounds like the problem is that the power conversion has a harmonic in the broadcast AM band, and it's getting into the audio system. That's poor design, but maybe they never tested it because AM radio is rarely used any more, and the inverter is not commonly used either.
You could try adding filtering on the DC side of the inverter. You would have to find the inverter (I think it's in the trunk, near one of the wheel wells), tap into the wiring harness, and add filter(s) appropriate for the load and frequency. The electrical system is complicated because everything is interconnected. I don't have time to design a filter for Toyota for free (they can hire me as a consultant and I will do it), but taking a guess I would try several hundred to 1000 uF of low ESR capacitors in parallel as close as possible to the inverter. You could also add some ceramic capacitors around 1-10uF each, but there might be too much inductance in the wiring from them to the inverter to be useful. You can get capacitors cheap from Digikey.com. Don't get stupid 1F capacitors that people use for audio systems. More capacitance is better, but large value capacitors do not work as well at high frequencies.
http://www.ti.com/lit/an/slta055/slta055.pdf
You could also try a common mode choke on the DC side of the inverter. Run the +12V and ground through it. It probably won't help much, but it probably won't hurt, either.
You could run a dedicated, fused, and shielded power cable from the battery to the inverter. I would use shielded twisted pair wire. Connect the shield and the negative wire to the negative terminal of the battery. Connect the positive wire, with an inline fuse, to the positive terminal of the battery. At the inverter end, ground the shield to the body as close as possible to the inverter. Connect the positive and negative wires to the inverter, and hope that the inverter has two terminals. Make sure the inverter is not connected to the body by any mounting fasteners or ground wires. If the case of the inverter is used as ground, electrically isolate the whole thing and connect the case to the negative wire of your new cable. You need something like 8-14 gauge shielded twisted pair cable, rated for automotive temperatures. Make sure it's lower gauge (thicker) than the existing wiring. I have no idea if you can buy a cable like that anywhere. It might not fix the problem, anyway.