Behind, but it's there
Right now they have 9 launches planned using Atlas 5. That's not going to get them a usable fleet of satellites. If they want to launch anything beyond that, Blue Origin must finish the BE-4: a rocket engine years behind schedule. (Either using them for Vulcan rockets or for the New Glenn).
From Amazon's perspective, those 9 launches are money wasted if BO doesn't come through. I'd be ecstatic if they did, but my money is on yet another delay from them.
If I had to guess, by the time Kuiper is launching enough satellites to matter, SpaceX will be doing the same thing using Starship at an order of magnitude lower cost. And I don't even think Starship will be doing real launches for years.
The answer is: nobody.
What made Starlink financially possible was SpaceX's lower launch costs and high capacity.
There's room for more satellites. Space is pretty big, and with some careful automated coordination we can avoid collisions. But someone else needs to first figure out how to cheaply launch a competing fleet, and I doubt anyone will come close for years.