Nel primo caso
To answer your question:
"mettere loro a disposizione" means "to make the group of people we are implicitly talking about availabe for the carrying out of some task at hand (e.g. "mettere a disposizione di "qualcuno"). The phrase must be followed by the preposition word "di", followed by a group of one or more words acting as a noun, which represent the object which is made available to such group of people. Anyways, I was misled by the OP's title and now I realize he was talking about something different: "i mezzi messi loro a disposizione" means "i mezzi messi a disposizione per loro", which means that
the means or tools required for carrying out the task at hand have been made available to someone or some group of people in order to satisfy the needs of the same people, or, possibly the needs of some other group of people.
"mettere a loro disposizione" means "
to make (something or several things) available to the group of people we are implicitly talking about".The phrase must be followed by a group of one or more words acting as a noun, which represent the object which is made available to such group of people.
"mettere alla loro disposizione" is a more verbose-sounding phrase carrying the same meaning as the second. In practice I would use it if I were trying to make a catchy, slowly-articulated, and emphatic speech and wanted to put particular emphasis on the point I was trying to make. The first construct is shorter than this one and flows better in normal everyday speech.
In response to a comment saying that
"loro" can mean
"to them" after the word
"mettere", I decided to point out the following rather unfortunate thing (which I would't have had the guts to explain were it not for the fact that this newsgroup is filled with experts who understand the fine aspects of Italian language and culture):
With "mettere", the only way to get a "to them" meaning, is to use the words "mettere a", but then the only possibility is the rather vulgar (and hopefully not too shoking!) expression "gliel'ho messo su per il culo" which, is grammatically equivalent to the sentence "l'ho messo su per il culo a lui/lei/loro" (not to be shocked here, in this sentence the subject can be male or female, the object can be male, female, or a group, and the case where the object can be a group makes you realize, I hope, that this expression conveys the feeling abstractness: regardless of sexuality, you cannot stick something in more than one place at the same time, and, it is precisely this abstract aspect of this very expression which makes it just as likely from anyone towards anyonw irrespective of the geneder of the people involved or their number). OK, enough said for the vulgar side, the grammatical side, and the abstract side. Now the meaning of the expression, which is not vulgar but on the other hand rather cunning! and has no equivalent expression in English, so much so that the only way to translate this expression is: "I've managed to deceive or trick the person or people in question in order to get what I wanted out of them at their own expense or loss and I'm quite happy about it". Sounds terrible? Well, yes it is. While talking to a group of friends some people could make use of this expression to point out what they have done to the other party which could even be a political party disliked by the group; they could even have befriended the other party in a false manner, or, someone could also use this expression to describe what a false friend could have done to them after they found out the truth, in which case the attitude tends to be one of deep disappointment and resentment! So much for sexual gratification! :-D
Now I see the full example with the words put in context. The first meaning of "mettere loro a disposizione" I gave above, is the one in sentences of the form "mettere loro a disposizione di qualcuno" which is what I originally the OP meant but I overlooked the example that was given. In the sentence "Grazie ai mezzi messi loro a disposizione, hanno potuto portare avanti le loro ricerche." the meaning conveyed is the fact that "i mezzi sono messi a disposizione per loro", hence we are talking about "mettere per" and not "mettere a" (because in English "to" corresponds to both "a" and "per", hence the confusion. Anyways, the expression appearing in the first case is archaic.
The second expression is the best, more natural choice.