Yes, you can shunt (short-circuit) the two signal conductors of the stereo cable and connect the output from the VCR to that cable with fear of damaging your equipment. This is a common quick fix for this type of situations.
frcake's comment about a signal splitter (by which I suppose it is meant an active device with a preamplifier) has some justification, as the output signal from the VCR will be split between the two input channels and so the input signal to each channel will be somewhat weaker. However you will probably still be able to compensate that with the input gain of your recorder device (I don't know the Elgato device, but normally there would be an audio input gain control somewhere in the accompanying capture software).
frcake's suggestion of recording in mono is also worthy to take into consideration. Creating the video with mono audio, as is suggested, is one possibility, but even if you want your final product to be stereo (e.g. because you're composing with stereo clips from other sources), you can always simply duplicate the mono audio clip from one channel to the other in your video post processing application.