Before we get into the whole where to find MAME ROMs online thing, we should say a thing or two about the legality of it all, as it might be of interest to some people, if not all.
First of all, one should remember that MAME is a piece of software that is absolutely free. The people who develop the source code wanted it to be a learning tool as much as anything else and they wanted everyone to be able to see it and make their own modifications to it. However, MAME does not contain any of the games in itself. It is merely a vessel which can be used to start ROMs which have to be obtained elsewhere.
You should remember that ROM, hard disk images and CDs are copyright material and that they are a property of certain companies. There are some ROMs which have been made public and which you can download perfectly legally from the official website of the MAME project.
However, these games are over several decades old and pretty much everyone is looking for something more. For those people, there are different ROM websites where the different MAME ROMs can be downloaded. Once again, these ROMs can be downloaded for free of charge 
and this is in fact the only way in which they can provide these ROMs as it would be illegal for them to sell them as they have been made by other people. In essence, it is all a bit murky when it comes to legality of downloading ROMs, but since many websites with ROMs have been operating perfectly above the board for years, it is safe to assume that there is nothing wrong with doing it.
Rom World is perhaps the best website to find your ROMs at. In addition to more than 30,000 different ROMs, at Rom World you can also find all the latest MAME and other emulators that have been developed recently. They also provide a community where you can exchange your experiences and much, much more. It truly is the best place to get your ROMs from. And not just that.