In the 1967 film, no explanation is offered by any of the SPECTRE agents as to how they are able to monitor their spacecraft closing in on the American vehicle in orbit, even though as the o/p correctly points out there is a live video feed displayed on a tv monitor screen in the SPECTRE control room showing it from a distant point-of-view that is clearly also in orbit.
I had always assumed that this is a vision feed from a SPECTRE communications satellite in a similar orbit. But I cannot find any other evidence in the film, apart from the obvious monitor pictures themselves.
In the plot, Blofeld is monitoring the capture of the American capsule because he intends to blow up his own vehicle immediately after the capture, in order to visibly destroy the American mission so as to cause a third world war. Hence he needs to be able to see when the capture occurs, to be sure of achieving this goal. Thus there is a legitimate reason why SPECTRE would have that monitoring capability, and so the plot is implying it is a SPECTRE satellite doing the monitoring. Bond simply doesn't waste any time on the point, having too many more pressing problems.
For those who don't recall the 1960s, there were in that decade many communications satellites in orbit: it was not a wild, science fiction idea. The space missions shown in the film were all potentially possible, as they all occurred in Earth orbit. The only element of fiction was the idea that anyone other than a major government could afford a manned spaceshot!