Will just a regular LED bulb work on my 2006 VTX 1300c without changing anything?
Yes, but you will have worse visible light than a halogen bulb... LEDs do not work well in reflectors designed for halogen bulbs. They will appear to be brighter (because they are), but they spread light all over the place, and less actual light in the correct pattern.
You are better of buying a www.vtxdaymaker.com
Daymaker (LEDs in a projector housing) WITH his adjustable bracket. It is well worth the money... Save your money and do it right the first time.
said, the bike will accept LED headlights without issue, it's the indicators that will rapidly flash if you don't change the indicator relay to a zero/no load like the one I posted.