Hey Adam;
Looks cool! A benchtop custom Luthier's Neck Vise. It looks good and solid overall, but yes, you need to add something to stop the racking/tilting of the moving jaw. There are several ways to do this.
One way is to use jacking screws, like you've seen on many of my routing fixtures. A pair of 6" long full-threaded bolts located down under the two vise screws, near the bottom. They come in from the back of the stationary jaw and thread through big T-nuts in the front face of the stationary jaw. The tip of the bolt presses against the bottom of the moving jaw, keeping it from tipping. For a fairly large vise like this, I'd recommend 5/16"-18 ot 3/8"-16 bolts. Your vise screws are pretty sturdy and can apply a few hundred pounds of force. The jacking screws are a simple solution which will work well. But they are somewhat annoying to have to re-adjust them every time you put a different size object in the vise.
A second solution is to use spacer blocks. Chunks of wood or metal that set down into the bottom of the jaws, under the vise screws.
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Here's an example. This is a small, special purpose wooden vise that I made for my buffing bench. It's made to clamp onto the 3/4" square aluminum tubes that I use for my body and neck holders.
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The vise is very simple construction, two big blocks of 8/4 maple and a rectangle of plywood. The left block is screwed down to the plywood base. The right block slides freely. The vise screw is a chunk of 1/4-20 threaded rod with an aluminum T-handle. There's a 1/4-20 T-nut set into the back side of the moving jaw.
Between the jaws, down at the bottom, is a chunk of 3/4" aluminum. It's just resting in there, not a permanent part. It's the same width as the part I'm normally clamping in the vise, the painting holder. That spacer block keeps the moving jaw from tipping. The vise can clamp very tightly.
You can put spacer blocks like this in your vise. For example, to clamp that tapered neck template, all you need is a scrap of wood of the same width and taper. Set it down between the jaws, at the bottom. Add some pins or tabs or something so it doesn't fall out the bottom. Tighten the vise screws, and it will clamp hard, without the jaw tilting.
The idea is that you have a small collection of spacer blocks to match the sizes of things that you commonly clamp in the vise. Set them in there and lift them out; change in seconds.
You can also combine the idea of the spacer blocks with the jacking screws. Have a few spacer blocks for the common size things, and/or adjust in the jacking screws for the oddball things.
The fancy way to solve the tilting-jaw problem is to use sliding bars, Common woodworking vises use two round steel bars on either side of the vise screw, The bars are rigidly attached to the moving jaw, and slide in bushings in the stationary jaw. Very strong, but more complicated to make up for your own vise.
Here's a simpler way to do it using rectangular aluminum bars.
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This is a special fixture that I made up to hold straight neck-lam strips while I pass them through my SPRP machine. It's a long, low profile vise, similar to yours. The right side jaw is stationary, screwed down to the base. The left side jaw moves, sliding on the base, and is clamped by three T-handle screws. T-nuts on the back side of the stationary jaw.
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Here's the moving jaw pulled out. There are two 1/4" x 1" aluminum bars rigidly attached to the moving jaw. They keep the jaw from tilting as the vise is tightened. There are also two guide pins with springs. They cause the jaws to open by themselves when the screws are loosened. A time saver in production.
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Here's the moving jaw flipped upside-down. The bar needs to be rigidly attached to the jaw. I extended the bar through the jaw, bolted to the two MDF blocks. The whole bottom surface of the bar rests on the base.
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At the stationary jaw side, the bar slides between these two fixed blocks, I've allowed a small gap on either side of the bar, between the blocks. This allows the moving jaw to go a little bit out of parallel, if I want to clamp a strip that's tapered. But the plywood block between the MDF blocks sits down on top of the bar, no real clearance, a nice sliding fit.
You could do something like this to your vise. It's more work to build, but there's no extra adjusting or swapping blocks.