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According to

https://elderscrolls.fandom.com/wiki/Yngol_Barrow

The area contains a puzzle door that requires the coral dragon claw to open, which can be found either from Birna in Winterhold or within the barrow itself, located on a pedestal past a puzzle room. If one obtains the quest to retrieve the Helm of Winterhold from this destination before talking to Birna and obtaining the coral dragon claw from her, and instead obtain the claw from within the barrow, then one can still return to Birna and receive the second claw for a modest price. However, that will not start a second quest, and returning to the barrow at that point shows things are exactly as they were left.

Well....

I passed the puzzle room with the snake in the grass, whale in the water, and eagle in the sun thingy.

This is the pedestal

enter image description here

I didn't get the claw from Birna so I should get that from that pedestal right? No.

I met Birna, and no coral dragonstone from her either :(

enter image description here

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If you go to the barrow before going to see Birna, is should be in the barrow. However, if you go to Birna first and do not take the claw once offered [e.g. if you missed it], it will disappear from the barrow.

In order to obtain it from Birna in the first place, you have to ask her if she has ever had a bad deal in her business:

If you ask her if she has ever had a bad deal after asking about her business, she will explain how she came to possess a seemingly useless dragon claw.

https://en.uesp.net/wiki/Skyrim:Birna

If you didn't take the claw from Birna, after it was offered it will not be on the pedestal either.

Once you've offered to buy the coral dragon claw from Birna (even if you didn't actually do so), the one on the pedestal in Yngol Barrow will disappear.

https://en.uesp.net/wiki/Skyrim:Yngol_Barrow

It seems to me that what most likely happened, if it's not in the barrow, is that somehow you missed the chance to purchase it from Birna when you spoke to her. That means that you either have to go back to an earlier save or forego the quest.

You could always go back to Birna and go through all dialogue options with her and see if you missed the offer, of course looking for the option about the bad deal.

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