My company offers a Simple IRA where they match 3% to what I put. I can only put in a max of $16,000 per year.
I'm guessing they match you contributions up to a maximum 3% of your salary, which is probably less than 16k (if you make $533k then congrats!!). It's still fine to max it out, but at some point you're not getting any more benefit from the company match.
Is Vanguard one of those companies that can help me out?
Yes - Vanguard is one of many large brokers that would be more than happy to help you invest your extra money.
Does Vanguard S&P 500 ETF qualify as an index fund that 'mirrors' the stock market
The S&P 500 Index represents the 500 largest public US companies, so it is a very good (but not complete) representation of "the market". Vanguard is one of many companies that offers an ETF that "tracks" (meaning is makes or loses value in step with) that index. If you see on the news that the "S&P is up 2%", that means that your ETF also gained 2% in value that day. There are ETFs that track many other indexes as well, but the S&P 500 is one of the most common.
It is a perfectly good place to invest money if you are not comfortable picking individual company stocks.
However, as you learn more, you may find that you are not comfortable with the level of risk that even a broad index fund gives you. It can lose money as "the market" goes down. You may find that you want to mix that ETF with other ETFs that are less volatile, meaning they will not lose as much, but they won't gain as much either.
Vanguard and other brokers have resources to help you decide what the appropriate level of risk is for your age, goals, temperment, etc.
Bottom line, talk to someone at Vanguard and maybe a few other brokers. You might even look for a local "wealth management" agent if you prefer to work with someone in person. It can be a cutthroat business, so stay away from those that feel like they're trying to "sell" you something (like insurance or annuities) and find one that is more interested in meeting your goals. They will all take a cut of your investments - some more than others - but that's the price to pay for expert advice.