I have Erbs Palsy causing my left arm to be shorter and weaker than my right arm. Can the handle bars of a road bike be adjusted to accommodate this or is there another resolution I can consider?
A solution will very much depend on exactly how much shorter and weaker you right arm is.
If the difference is small, you could adjust the handlebars to be correct for your left arm by bringing them higher and further rearward by fitting a shorter stem with greater rise. This of course compromises the position of your right arm to some degree.
If the difference is too great, I think you would need to look at a custom built handlebar. This could possibly be made by cutting a bar 5-10cm to the left of the the stem clamping area, and adding a block that clamps the two together and moves the left side up and back.
If you are using a road bike drop handlebar, possibly you could adjust the bar position so that you ride with you left hand on the hood position, and right hand on the drop position.
Presumably another concern is having enough strength in your left hand to operate the front brake and shifter effectively. The majority of braking should be performed with the front brake so a stronger right hand is preferable to a weaker left hand.
If you can't operate the front shifter with your left hand you could of course fit a one-by drivetrain with only a rear derailleur and shifter.
For a small difference in reach, it could be a simple as adjusting the position of the hoods on your handlebar such that the left one sits a little higher and further back. It's a change that will often look small, but it adjusts the angle of the lever at the same time, which amplifies the change too.
I used to regularly see a cyclist with one noticeably thinner and shorter arm. He used a normal road bike and just had his weaker arm on the top of the bars all the time, while using all positions with the right one. He also had an extra brake lever on the right. He seemed to do quite well, even climbed very steep hills, but I've never talked to him.
If the difference isn't too great, you can try simply rotating the stem so that the bars will be slightly askew.