Can the vaccines cause my 2-month-old to run a fever?

Yes. Yes - vaccines cause an immune response to what is contained in the vaccine. This may, and may not, include having a fever. Usually, a fever related to vaccines will last up to 3-4 days and not be terribly severe. Any fever that is very high, or any time the child is acting very ill, even after vaccines, you should discuss the situation with your doctor.
Yes. Your baby may run a fever and be irritable for about 1 day after the shots. Your baby may also have some soreness, redness, and swelling in the area where the shots were given.