There's two problems with injecting yourself with a vaccine
One is the vaccine itself - basically it's something that is MEANT to create an immune reaction, and if you have an immune reaction under your skin, that's inflammation, ie. redness, heat, swelling, pain.
Second is that the needle is likely to have been used in an animal, and so is not likely to be clean, so may introduce dirt/bacteria etc., which can cause an infection in the bit that was injected, which can become an abscess.
Anything like this in the hand area, where injections can end up very close to bone, joints and tendons is VERY serious. Inflammation or infection in your joints, tendons etc, could mean loss of use of hands.
I would be going to see the doctor first thing on monday at the very least, if you are struggling to move your fingers, I would strongly recommend A&E.