I've been invited to join a current campaign, starting level is somewhere between 8-10. When I asked what the party needed, the response was "Well, they don't really have a caster. Of any kind."

So, given that I prefer physical combat to spell duels, what should I make? I'm debating between a straight cleric + PrC, a Ruby Knight Vindicator (probably capping out at 8th level spells), or a Psychic Warrior or Ardent (maybe Tashalatora style).

No caster of any kind makes me immediately think you need a wizard, but since that isn't your thing, I'd go with a domain-focused cleric, Sovereign Speaker/Contemplative/whatever else you need. Massive list-expansion can help you pull some wizard weight decently. Actually, Elf isn't a particularly suck choice here, since SotMI is pretty pro.

If your party lacks any casters, at higher levels you really want something with a big three-sized list and full access. As such, I recommend against RKV, the CL problems are too much, not to mention its main draw actually doesn't work. Either way, with the things your considering, I have to recommend CC+PrC.

Where do I find that? From what I've been able to find, it's from Dragon Mag, and I don't know if that's available - or even what the difference is. I'll look into those. SS is Magic of Eberron, and Contemplative & SotMI are both CD, correct?

It's a build type, no different from Mailman or Batman wizard.
You appear to be thinking about the Evangelist cleric. It's awesome, but Dragon material, sadly.Faiths of Eberron, actually, unless I'm mistaken, but besides that yep.

Ok, so a domain-focused cleric is one that tries to get access to as many domains as possible? I can live with that.Well... crap. I have MoE, but not Faiths.

Isn't there a class in Planar Handbook (or Manual of the Planes?) that also grants access to an extra domain? Also, is there a domain that allows access to the wizard/sorcerer list?

No, it's one that tries to access as many good domains as possible.
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If there is, it sucks. The alignment and element domains are almost universally terrible.Sort of. Divine Magician, from Complete Mage. Lose a domain, get some wizard spells from limited schools. It's a pretty good deal a lot of the time.

Ok, so assume Corellon, since I like SotMi (at least the first 5 levels!). His domains are Chaos, Good, Magic, Protection, War, Community, Creation, Divination, Elf, Exorcism, Glory, Mind, Mysticism, Summoning

Which ones are the good ones? I assume that Knowledge (from Cloistered) and Travel (from SotMI) are going to be traded for Devotions, right?It grants an extra domain from your allowed list, not alignment or element.Sweet!

Corellon is entirely disconnected from SotMI.
Protection is okay, although it's Devotion is pretty good, IIRC, War is pretty bad but the real draw is the Holy Warrior feat, which is hella kickass, Creation is okay, Elf is for the bonus feat, that's all that needs to be said, Glory depends entirely on your campaign, Mind is pretty good, and I didn't think Corellon got Summoner, but it's quite a good domain. Oh, and Magic is so good I don't think I need to go much further.

That said, I really don't like Corellon.
Knowledge? Yeah, the domain kinda sucks, but Travel? I personally like the Travel Domain. The amazing spell list and very good power generally means I value it far more than the Devotion.
Oh, yeah, that class. It sucks. You're a Sovereign Speaker if you're doing this right, and they'll have all the domains you'll need and then some. You can't afford the CL loss like that. I mean, that class has some uses, sure, but it's dramatically inferior to most other domain PrCs. Go Contemplative instead.