Are there types of glue that are designed to be dissolved easily?
"Wood glues" can have several different compositions, chances are if the ones you are working with respond to vinegar they are either protein based or PVA.  In the protein case the solvation is likely accomplished by the vinegar protonating the polymer chains, and therefore changing their attraction to each other. In the case of PVA based glues, vinegar is likely contributing to acid catalyzed hydrolysis.  The softening of cyanoacrylate (super glue) with acetone is a more traditional dissolving mechanic, as no chemical changes are being made.

That being said, all three mechanisms mentioned above can be pretty well represented in the glue world with different polymers.  Based on what you consider cheap solvents, you have a few options. Cyanoacrylate (superglues), some UV-cured adhesives (which are frequently very well crosslinked and strong), PVC, and many others are all softenable and removable by acetone

Additionally, anything that can be taken out by acetone will be solvated in MEK (methyl ethyl ketone).  MEK is less water miscible and more aggressive though, which may be relevant to your needs.

Silicone adhesives are great for flexible applications, but frequently need stronger organic solvents (like toluene) to swell it and then it can be mechanically removed.  There are also commercially available silicone removers that appear to use acidic or basic bond cleavage to break up the adhesive.

Rubber cement is another flexible and relatively strong bonding adhesive mainly composed of solvated polymers, drying and tackifying by solvent evaporation.  Reintroducing the solvents (many commercially available paint thinners should work) will soften the adhesive.

As an aside note, depending on the materials you are bonding together, heat and cold can also be used to weaken joints and destabilize the adhesive bond separately or in tandem with chemical treatments.