I would definitely call your primary care doctor
's office (or the office of which ever doctor gave you this shot) to discuss this issue. Although it is not very common, vaccines are an invasive procedure and there is a small chance that an skin infection could begin at the site where the needle went in.
Typical signs of a skin infection, or cellulitis as it is called in medical terminology, would include pain, redness, swelling, and warmth surrounding the site of the injection and spreading outward. The description you give of the red sore bumps extending down the arm is not perfectly consistent with this type of skin infection, but I think it is worrisome enough to get it checked out.
Your primary care doctor will be able to take a look and help you decide what to do. If there is evidence of a skin infection, you will probably need to take an oral prescription antibiotic, which your doctor will be able to provide. They will also be able to follow-up with you to make sure that the infection is clearing up.
Please call your doctor's office right away, and I hope that this issue clears up quickly!