I have a situation where the bike has (OEM) 8-speed Shimano Claris shifters with the cables coming out the side which interferes with the handlebar bag. (Photo below). SRAM doesn't list any 8-speed brifters at all, and Campy 8speed ergos are only available (second hand? genuine?) from Ebay. I'm looking specifically for brifters with internal cable routing - bar-end shifters are a fallback option.
There are two ways to do it.
One is use Microshift SB-R482 levers. (Sorry, I got this wrong because the image is a little tricky and it comes off a link from their front page that says "inner route cables." These have shift cable routing much like STIs with external routing.)
The other is use 10 speed Ergo levers, or 9 speed Ergo levers with the Hubbub/Shimergo mod, to shift Shimano 8 with a Shimano derailer and cassette, as outlined here. Or if you happen to find an 8-speed Ergo, you can install a 9-speed or 10-speed ratchet ring into it.
Interference with handlebar bags is a big reason why the Shimergo/Hubbub trick became a thing. You can also do the same trick to make 10-speed Ergo work with 9-speed Shimano.
SRAM started at 10 and never went backwards, and Shimano also has never made 8-speed STI with under-tape shift routing.
The new Claris, called R2000, has internally routed brake and shifter cables, an improvement on a previous version, that still sported the externally routed shifting cables.
It is still 8 speed group, so in line with the op's wishes.
dealers manual, page 13, how to install shifter cable: http://si.shimano.com/pdfs/dm/DM-RBSL001-03-ENG.pdf
Although your pocketbook won't like this solution, the Alfine Di2 wireless brifter from Shimano does have an 8-speed option. It's not as much an internal [shifter] cable solution as a wireless shifter and internal brake cable solution.
As commenters have pointed out, the Alfine is for internal geared hubs. You might be able to bodge a solution with their Dura-Ace wireless brifters but it'll be similarly expensive and may not work.