Is there some easy way to share SpaceChem save data between computers

Bonders are moved in the same way that instructions are moved after placement - usually this is by clicking on them, holding and dragging them across the field.
(If you are somehow moving instructions in a different way, try that as well)

Note that you cannot use a drag-box selection (as you can to pick up multiple instructions), these ignore reactor features such as bonders.

If you can't move bonders in the same was as instructions (and maybe you can't move instructions either?), it sounds like you've hit a bug of some type.

As William suggests in a comment, do you have anything unusual with respect to the set up of your input devices? And, have you tried moving the bonders (or different types of reactor features) on different levels in case it's a level specific problem?

The folders that need linking are: (for Windows)

C:\Documents and Settings\<username>\Local Settings\Application Data\Zachtronics Industries\SpaceChem

...and: (for Linux and OS X)

~/.local/share/Zachtronics Industries/SpaceChem