Totemists only receive light armor proficiency. Like any other character, if they wear armor they are not proficient in, they take penalties. None of these penalties directly affect their meldshaping, however (and arcane spell failure does not apply to meldshapers). Totemists can gain proficiency in a given armor, like any other character, by multiclassing or taking the appropriate proficiency feats.
Meldshapers in general have some concern about magic items and soulmelds: a soulmeld that is bound (not merely shaped, but bound) to a chakra suppresses the magic on any item in the same item slot. For magical armor, this is the soul chakra—which totemists don’t get (barring an Epic feat). But for incarnates, who open the soul chakra at 19th, taking the Split Chakra feat at 18th (or earlier, if more convenient) to allow simultaneous usage of magical armor and soulmelds bound to the soul chakra is recommended.
Note that even for an incarnate without the Split Chakra feat, a soulmeld bound to the soul chakra would only suppress the magic on the armor: its mundane properties, including those from any special materials used in its construction, would still function fine.
The only thing specific to a totemist that has a problem with armor is the fast movement feature offered by the halfling racial substitution levels, which only functions in no or light armor. Even a halfling totemist can choose to simply not take that level if they wish to wear heavier armor, though.
Regarding meldshaping and allowing a totemist when you are not familiar with the material, it is an excellent system, but a fairly complex one. There is quite enough room for a person to legitimately misunderstand how things work—and then when you have a player known to sometimes bend rules in his favor, that means it’s pretty easy to do so with plausible deniability. In fact, much of this problem is due to Magic of Incarnum itself—the book is not well organized or laid out, and it makes the system more difficult to understand than it needs to be.
Instead, I recommend that you check out Sinfire Titan’s Incarnum Handbook. It’s written from the perspective of someone trying to optimize these classes, but it spends a fair amount of time also just explaining how the system works, since Magic of Incarnum is so troublesome. It’s also a much quicker read (particularly if you skim all the various options and just read the basics), and much clearer. And since it’s written aiming to optimize these classes, it also gives you a decent sense of how much they can do, on the high end. Overall, these classes are fairly well average in their power, relative to 3.5 as a whole (stronger than most pure-mundane classes, but falling well short of true spellcasters), and they’re great fun, so I heartily recommend them—once you have a decent grasp on how the system works.