Inverters generally don't have a battery. They only convert the electricity output of the truck (DC) to household power (AC). If the inverter is hardwired to your truck (it should be if it's pulling more than about 120w) it should work whether the truck is running or not, however heavy loads like running a microwave will be easier on the batteries if the engine is running. I have never had an inverter that had a battery or needed to be charged, so I'm not sure how to answer. What inverter do you have, specifically? Model number, etc.
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Most inverters have a battery charging indicator and if you don't have a battery it always shows charging. That's how the few I've seen work anyway.
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Help I have a xantrex power inverter with supposedly a new battery. The thing won't stay charged it always has a yellow light. My question is. Does it charge better off or on while I'm driving? This thing is driving me nuts. Thank you