The Tavern

I am intersted in the viability of using ranged weapons as a character's primary killdudes without worshiping an ammo god. Here are my non-expert thoughts on this:

Throwing: If you are Og/Tr, then once you get to Depths and Vaults you'll have enough large rocks. Otherwise, even with an ammo god it's hard to find enough javelins. Orc gives tomahawks, and Shoals is really helpful.

Slings: Great early, stones are plentiful and bullets common enough. Shoals again helps with ammo, otherwise you run out of bullets at about the same time that sling damage falls off in effectiveness. If you can get a boatload of slaying, though, even stones are effective through Zot and you'll never run out of those.

Bows: Consistent sources of arrows throughout the game after mid dungeon starts spawning centaurs. Snake often has a lot of arrows. Longbow is really good.

Crossbows: Once You get to Vaults, you have enough for the rest of the game. Snake can help, too. Otherwise, hit or miss on finding enough bolts.

Which gods other than Trog and Oka make good gods for this playstyle, if any?

For throwing, stones have an amazing damage output at high skill, and can be used at low skill when all other ammo is gone. You will never run out of stones. Again for throwing, Chei is good for increasing your damage (and that is particularly true when you can lob large rocks); his speed penalty is not so much of a problem, as you can kill things at LOS. Your DEX-based stealth will also help to control what you are fighting, even with heavy armour.
Makhleb is also good, as his minor destruction helps to bridge ammo-less stretches of the early game; his passive regeneration on killings will keep you afloat in long fights with big crowds.

Fedhas, because you can cheese the hell out of allies and plants. Also, because Fedhas.

Dith's shadow attack helps missile damage output, though I don't remember how strong that effect is.

Dith. the shadow mimic is like a 1.5X speed finesse that stacks with haste and is on all the time.

I realized Dith also saves ammo because the shadow mimic kills stuff quicker using no ammo.

I am not sure this is correct. I am sure it was not so before Throwing change.
From my pure throwing game (0.15):

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Having Shoals increases the javelin supply a lot.

If it is a response for my game, you can see from action table that I didn't have any problems before Shoals either (entered it at XL 17). I just stopped using stones vs popcorn.