If you select two adjacent nodes on a path, you can move them closer together or further apart symmetrically using the arrow keys. Is there a similar keyboard shortcut, function, or option for objects that replicates this functionality?

I realize that the Align and Distribute window has Remove Overlaps, which can push objects equally apart from a central point, but that only offers a partial solution. If the gap you specify is smaller than the gap between the two objects, Remove Overlaps does nothing.

Currently my workaround is to move one object manually toward or away from the other until the gap is the correct size, then to vertically align the objects (grouped or selected together) to whatever represents the center point. I'm just wondering if there is an easier, built-in function that accomplishes this.

Welcome aboard!

You can try to use the transformation panel (Shift+Ctrl+M) for that.
Select all objects, disable proportional scaling,
scale objects together down with a percentage only in horizontal/vertical direction, then scale up those,
BUT this time with the apply transformation on each individual object option enabled.

There are some limitations though with the percentage input.
Without sticking to pairs like 50%-200%, or 80%-125%, the result won't be accurate.

Thank you for your solution! I believe I followed the directions correctly, and while it certainly moves objects toward and away from one another, I found it difficult to make the objects meet up exactly on their line of symmetry.

I did discover this, though. After one or two iterations (this depends on how far apart they are to begin with), the objects will probably overlap over their line of symmetry; at that point Remove Overlaps set at 0 will push them apart so they abut their line of symmetry.

That said, an easier solution might be to manually move each object an equal amount (using Shift/Alt + left/right/up/down) towards the line of symmetry until the gap between them is smaller than required, then apply Remove Overlaps.