Having seen the photo of the entire piece I'm with Coley, fill the gaps with slivers of Oak as best you can. With care that could look entirely convincing on the
sides of each individual tail, but it won't be quite as neat plugging any gaps on the
end of a tail. You'll also have to be very careful dealing with glue squeeze out from your Oak slivers and patches, any PVA left in the open pores of Oak will be pretty obvious under an oil based finish. In squeeze out critical applications I prefer one of the liquid hide glues over PVA (Titebond do one that's available from Axminster and Workshop Heaven have just begun stocking "Old Brown Glue"), these clean up with hot water far better than PVA.
After doing the best job I could with slivers of Oak I'd then look to fill any remaining gaps with a coloured wax. These patching waxes fall into two camps, there's soft waxes like these,
https://www.wood-finishes-direct.com/pr ... gJOevD_BwE
Then there are hard wax filler sticks like these,
https://www.glenwooddecoratingsupplies. ... gLyNPD_BwE
The hard wax sticks are much better but are more difficult to apply. If you decide to go this route let me know and I can help you with the process. In fact if you want some small samples of Oak coloured soft and hard waxes PM me and I'll post you some pieces to experiment with.
There is a face frame option, but looking at the photo of the entire piece it's clear it would be quite a bit of extra work and it could look artificial where it's stuck on the front of the legs, plus it would be awkward to get the necessary precise fit around the drawer openings.