I'm trying to make a character using the kobold option presented in Volos but it's not showing up for me even though the other Volo options are. Is there any way to remedy this? I have a character saved that has this option selected so I can copy it but I'd rather not have to do that when I make a new character. 

Do you have Legacy Content enabled on the racial selection page?

EDIT: It's a slider just to the right of the Racial Filter dropdown menu at the top portion of the page. The tab should be slid to the right, and the icon then turn red.

I do have legacy content enabled and all of the other legacy variants are there except the kobold. 

Is this one of the copied versions from the character you have saved with that option, and/or do you already have Kobold selected as the race? (I have done this more times than I care to admit, lol)

Also, do you personally own the content for your own access, or was it granted through content sharing?

I do personally own the content. It only shows up as an option under the copy I've made of my original kobold. If I go to make a new character, the option is no longer there. 

Once I change to anything other than kobold the option to change it back is no longer there. The other legacy options from volos are though

Make sure your Account Entitlements are in sync: (https://www.dndbeyond.com/account/marketplace).