Are there any recommendations for where I can buy an inexpensive bike stand clamp? I do not know what they are exactly called but the actual part that the bicycle is held onto in a work stand.
If you are looking for just the clamp, and want something produced commercially you are going to have a hard time finding something inexpensive (at least by my budget). A Google search for Park Repair Clamp turns up a number of options but they are generally $100 and up. You can pick up a decent stand for that much.
The best deal I could find was a brand I'd never heard of:
Your other option would be to build your own stand:
Hopefully you get the idea...
If the clamp in Gary's answer is too pricey, it's time to start looking at other options than an actual clamp.
Even though it's nowhere nearly as flexible as a proper repair stand, a wall mount bike rack works quite wall as a repair stand. You screw one of these into the wall at shoulder height (or whatever height you like) and put the bike in it, drivetrain-side out. You now have a decent repair stand - the pedals will rotate freely if there's nothing on the wall to prevent it. (Check the depth of the stand before buying, though.) These go for about $25 at most hardware stores.
Also, a car rack will do a decent job as a repair stand. These generally cost almost as much as that stand, so do this if you already have a car rack or if you find one really cheap.
The pedals often do not rotate freely on these, however, unless the bike is all the way out from the car. You want the sort of rack where the bikes hang from the rack, not the kind where the bikes stand on top of the rack. Look for a 3-bike rack or more, those tend to protrude out from the car more. Working on your bike when it's on your car can scratch your car or get it dirty, so this isn't an option on that bright yellow sports car.
Floor stands also work fairly wall, but have many of the same limitations as car racks; they're almost always not deep enough for the pedals to rotate freely.
Find a decent pair of medium vice grips, and weld a pipe approximately 1-3/8 diameter by 3" length, (you'll have to cut into 2 equal halves by cutting down the 3" length), weld 1 of the the halves onto the upper and the other1to the the lower jaws of the vice grips perpendicular to the pliers. You attach to whatever piece of stand-off arm by welding, clamping, whatever. I built my bike stand from parts thrown away by local businesses. The lower stand is made from a collapsible P.A. monitor stand, that is adjustibe by width of the3 outrigger "feet", and adjusts to 5' high from26". The center section is made from a steel adjustable telescoping floor vacuum pipe. The top section is from a drum riser and several pieces of hardware/ clamps made for the drums, cymbals. I am currently working on some seatpost clamps, but am currently using handlebar end "antlers" wrapped with foam grips. The stand holds 2 mountain bikes and it's not even a bit strained.
Can you weld? Some of these homemade efforts work really well, and will last many decades longer than plasticky ones.
Finally here's what I made, usiog some old scaffolding cross braces. Theres an old concrete umbrella stand at the bottom of the pole, and all my tools can sit in the rear of the car for easier access. Only problem is the bike wobbles about if you're leaning on a spanner, but it provides easy access without bending down.