I am wanting to get a small mechanical arm that can move something back and forth if you inches via remote control. Where could I find something like this?

To my knowledge they do not sell generic or standard mechanical workings. You have to make yourself or find a mechanical engineer and have them design it, then maybe they can point you to a tool and die maker that will bring it life. There are tool making companies that you can hire to make factory tools for production in a factory. They usually do little side jobs as well. Remote controls can be found on eBay from China, only cost a few dollars.

Some tool and die makers that are retired, still work in their basement or garage. You just have to search and ask around. Technical schools is another place to look. The teachers sometimes work on projects on their own.

Mndude,
There are tons on Amazon.
Search for "mechanical arm remote" and scroll down all the search results.

I built such a thing in a close up case: https://m.youtube.com/watch?v=vvv_QwrMSvs

There is a robotics kits and all kid of parts for this type of parts the company is called pololu electronics in Las Vegas

Another place you might look is online hobby Shops. The servos used in remote control cars or airplanes could be perfect. There used to be a place called Tower Hobby. I don't know if they're still around.

Here is a link to a magician who makes remote controlled parts for magicians.
https://ideaalab.com/en/lanzador-electronico

Here's another source: Blue Point Engineering

http://www.bpesolutions.com/

Here is a linear moving part which is controlled wirelessly

https://www.amazon.com/ECO-WORTHY-Actuat......0XHDAFZC

If you are looking for a source for tons of gears and levers, etc., find an old NCR mechanical cash register and take it apart. It is all gears, cams motors, lever, pulleys, solenoids, etc. You would be amazed at how much is packed into one case.

Gary Salisbury
former Field Engineer for NCR (National Cash Register)

Vendors that deal in arduino, such as adafruit, can accomplish quite a bit. Relatively user friendly if you take the time to learn. A lot of people will write code for you if you describe what you want to accomplish.

The remote for the new light bulb trick Shine works well at a great distance. How could I use it to light a battery powered candle or other device? That must be possible, right?