Can I get a pilot's license following a vehicular homicide charge?
I'm assuming you're asking about training in the US because "vehicular homicide" is an American term, at least as far as I know. If that assumption is wrong, please update your question to include the country. You also didn't mention if you were only charged, or actually convicted.

Anyway, if we assume the US then you need TSA approval before flying solo (for citizens) or before starting flight training (for non-citizens). I have no idea how the TSA would view a vehicular homicide charge (or conviction), but obviously it's a potential issue.

But assuming you get TSA approval, you'll also need a medical certificate. Criminal charges or convictions shouldn't really matter for that, unless they involve drugs and alcohol (as you mentioned) or if the incident led to you needing long-term drug treatment, a major operation or psychological/psychiatric treatment. The FAA views psychiatric treatment in particular (including most counselling) very negatively, so if you had any that would almost certainly result in a lot of extra paperwork and potentially denial of your application.

Alternatively, you could go for a sport pilot license and fly an LSA without any medical if you have a valid driver's license. But if the FAA has ever denied a medical application from you, you can't use that option, so research and plan accordingly.

But if there are no medical issues the only other thing I can think of right now would be insurance. There's no legal requirement in the US to have any insurance in order to fly but most people who rent aircraft (including students) will have renters' insurance in case they damage the aircraft or injure someone or whatever. If your homicide charge resulted in a conviction, that could be a problem for an insurance company. Even if you don't get your own insurance, you might have to sign something for the flight school's insurers stating that you've never been convicted of a felony or motor vehicle charge (I have no idea, by the way, if vehicular homicide is a felony or not). I'm not saying that it would be impossible to get insurance, just that it might come up when/if you apply for it.

Finally, if you eventually want to get an ATP license then a felony conviction is a potential concern: see this question for more information.

Disclaimer: I'm not a lawyer or a flight instructor so don't take my word on anything important (or unimportant, for that matter). Legal issues are always tricky and for a real answer you should consult an aviation attorney or join AOPA and ask them.