I think that you can still say Channelers get their power from their god, as there are more than a dozen major gods and surely many minor ones in the Tolkien mythos. The problem (at least from my perspective) is that Tolkien seems to speak of religion rarely, and describe religious rituals/practices even more rarely. He seems rarely to mention things like churches and temples. So I'm not exactly sure how the gods were worshipped, other than the Elves singing songs in their honor.
There have been other replies that already refer to some crucial topics in this context. A very important one is to decide on how "canonical" you want your game to be, i.e. do you stick very closely to what Tolkien has written about his cosmology (and not only in the LotR but also the numerous additional works; especially the HoMe series) or do you follow it more loosely, allowing considerable leeway (and thus use the background more as a rough guideline)?
In the first case, Paladins - and Chaneling magic as defined by RM (and MERP!) - would not be existent for anyone aligned with the Valar. In a nutshell, the Valar are "only" the guardians of Arda and are theologically more like saints or angels rather than gods themselves. The most central and important commandment laid upon them by Eru is that they shall make Arda habitable for Erus Children (Elves, Men and Dwarves) but not interfere with them in a way such as influencing their decisions and freedom by a display of their - clearly - superior power and majesty. Naturally, such a commandment includes not to "lend" them any power they do not possess innately. Since "Channeling" is defined as exactly this - lending power of a god to his worshippers/devotees to work "magic", it is only possible for the "evil" guys who do not care about Eru's commandments. And even then, this would sap at their own innate power and strength if done by a lot of people over extended periods of time. Melkor/Morgoth might be prone to this (as he gave most of his power into his creations like orcs, trolls and dragons anyway), but Sauron who was more wiser kept his innate power more to himself rather than disperse it.
This partially answers also your other question: Tolkien refers not to temples, priests and churches among his "good" people since they have nothing of this sort! He discusses this in context with the Númenóreans/Dúnedain (synonymous terms): In Númenor, the king was the only priest, as only he was allowed to make the spiritual contact with Eru. And this open "service" was done only on the holy days upon the peak of the Meneltarma. Beyond this, the Númenóreans had nothing like any organized church, clergy etc.
The first (and only) time a temple was built in Númenor was when Sauron seduced them to Morgoth-worship (with Sauron as High-Priest) which ultimately led to disaster.
Tolkien's deep catholic faith prevented him from creating any religion contradicting his devout christian belief even in an imagined world. In a way, he devised it as "mythologic" past of our own world when Christ has not appeared yet and all the "good" people in this story have to live without the promise of salvation he has taught.
But OTOH, if you do not plan to stick this closely to Tolkien's cosmological ideas, there have been some quite useful ideas in this thread already
Cheers
Tolwen